6:45, Day 3, NUS high school
今天是繳交written report的日子，我們要做的事就是把昨晚做出來的內容文字化交出去而已。有電腦，有MS word，我們以為要打一份報告十分容易，其實不然。
一台手提電腦，缺掉滑鼠，其機動性就好像減了一半。特別是針對給定的地圖上畫畫作出解釋，這對我們難度十分高，要用手提電腦的觸控版實在太難了。
不過，要說到港人還有甚麼優點的話，那就是夠快了。答案是在昨晚figure out，至於要怎樣下筆，則從來沒有想過。所謂的報告應該有格式無格式，還是好像公開試那樣寫上題號就能作答？我也不知道。但真正坐在電腦前方，身邊的一切就在這一瞬間凝固掉，身處一間沒冷氣室溫三十五度的房間也好，隔壁的中國隊無聊得在跳舞也好，我們唯一看到就是眼前的文件。 (這倒有點像HKMO, 聖賢之類的團體賽，能幾個人圍在一起計一題數的情況真少見)
於是，很快地，這份報告就像一份第一堂收的功課一樣被趕起了。沒有多餘的修飾，只有簡陋的說明圖；沒有華麗的算式，只有簡單直接的解釋。
……
正午是報告的截數時間，從十時開始趕工，到最後我們早了三十分鐘交出來，可說是不錯的速度了。不過，書面報告之後，還有一項更剌激的，就是口述報告。
在第四日早上，我們會隨機被分到四位評審，我們要分四次單獨的解釋我們的報告。在這種問題下，排除窮舉外幾乎沒有「絕對正確」的答案，因為這部分最刺激的地方在於評審有權隨時打斷你的發言來質問你(原則上，Q&Asection是最後五分鐘，但事實上顯然不是)。
此時我們開始預備簡報，問題是，我們的解答中，證明了幾個直觀排列「是開店的必然法則」來證明我們的答案。很可惜，當地圖越來越複雜時，一間店身處於某座標時不能同時滿足所有的必然法則，在「開店開小化」下我們顯然不能滿足所有法則，那在直觀排列下的可能答案就不止一個了。窮舉顯然不是很好的辦法。
晚上我們暫停了手上的工作，選擇了跟各國選手比試一下－－當然不是數學，我們選擇了另一種全場通用的語言－－體育作為切磋的部分。雖說是比賽途中，但每個人就輕鬆得像在學校跟隊打一樣，我們就這樣渡過了一晚。
順帶一提，這天晚上的食物是類似炒年糕的東西，再配以炸魷魚(煎炸炒，很地道的煮法？)，但最神奇的是前幾天桃紅色的賓治被稀釋成淺紅色，據說有人服用後腸胃不適……
不過把雜果賓治撇開後，能夠在同一個地方幾天內吃到各國的美食也算是不錯……(你說甚麼？香港也能吃到上述各國地道食物？這點我也不得不承認，但我在這裡是衷心覺得他們安排不錯的呢。)
這晚沒有英超，我們卻再次聚到同一開房裡開通宵，將那萬惡的簡報預備好，然後爭取為數不多的寶貴休息時間。
……
Friday, 31 December 2010
Wednesday, 29 December 2010
Economics : Inflation
Inflation
It’s the persistent increase in general price level, while deflation is the persistent decrease in general price level. The followings are not included as inflation:
1) Temporary rise in price level (e.g. during Festival, weather)
2) Once-and-for-all (only one time) rise in general price level (e.g. imposes of tax)
3) Persistent rise in the price of a single good
Inflation or deflation rate = %ΔPI = (PI_{2}-PI_{1})/PI_{1} * 100%
Nominal value (money value) is the value (in terms of money) in the period. Larger general price level implies smaller purchasing power per unit of nominal value.
Real value implies the value in terms on money in the base period. It implies purchasing power directly. Real and nominal values are related by:
Real value = Nominal value in a specified period/PI in that period * 100
Nominal interest rate (n) is the rate of change in nominal value of an asset over a specified period of time. Real interest rate (r) is the rate of change in real value of an asset over a specified period of time. The inflation rate (i) is given by: i = n – r.
Usually the inflation rate in future is unknown at present, so there will be a n expected real interest rate (r^{e}) : r^{e} = n – i^{e}, where i^{e} is the expected inflation rate.
Also, the actual real interest rate r^{a} can be given by r^{a} = n - i^{a} where i^{a} is the actual inflation rate.
When the actual change in price level differs from expected, there will be a redistributive effect between providers and recipients of fixed future payments.
- Anticipated change in price level is expected (correctly).
_{- }Unanticipated change in price level is unexpected or expected incorrectly._{}
_{- }Under inflation, providers of fixed future payments (borrower) gain while the recipients (lender) lose. It’s because the interest rate of fixed future payment is bonded by expected inflation rate. When unanticipated inflation exist, the value of returning sum is less than expected in real terms (i^{a} > i^{e}, r^{a} = n - i^{a} < n - i^{e} = r^{e})._{}
_{- }Oppositely, under deflation, providers of fixed future payments (borrower) lose while the recipients (lender) gain._{}
Tuesday, 28 December 2010
Economics: National Income statistic II
Nominal GDP VS Real GDP
Nominal GDP (GDP at current market prices) is the GDP measures at market prices of the current period, i.e., GDP at current market prices = ΣP_{c}Q_{c}, where P_{c} is the current market prices.
Real GDP (GDP at constant market prices) is the GDP measures at market prices of a specified base period chosen, i.e., GDP at constant market prices = ΣP_{b}Q_{c}, where P_{b} is the current market prices.
It can be measured by revaluing the market price in base period or deflating the nominal GDP by a certain price indexes. Real GDP is a better measure of GDP since the effect of change in price level is eliminated so that change in real GDP implies change in aggregate output.
Per capita GDP/consumption = GDP/consumption divided by population, mainly used to show the living standard.
Full-employment GDP (Potential GDP) is a theoretical GDP when all resources are used efficiently. It eliminates the effect of misuse of resource and unemployment problem.
Demand side factor affecting actual GDP:
- C: increase in income/tax allowance or decrease in salaries rate increases consumption
- I: decrease in profit tax rate/interest rate increases investment
- Change in G, X, M affects GDP directly.
Supply factor affecting the potential GDP: quantity/quality increase in factors of production, technology, favourable economic policies, political stability, etc.
Uses of national income statistics
1) Assess economic performance of an economy
2) Reflect economic welfare (living standard) of a economy
3) Facilitate international comparison
4) Provide information for government to formulate economic policies, e.g. value-added of different industries shows that which sectors need more support.
5) Provide information for firm to make production and investment plans.
Limitations/weaknesses of national income statistics
1) Value of some unpaid services are excluded – underestimating economic welfare
2) Inaccurate estimation of illegal, unreported and non-marketed production
3) Value of leisure is excluded (more leisure time under the same GDP is preferred)
4) Undesirable effect of production is excluded (e.g. damaging the environment)
5) Change in general price level is ignored in nominal GDP
6) Population size is ignored in nominal GDP
7) Composition of output is ignored (e.g. only C, G relates to the economic welfare)
8) Income inequality not shown
9) Market value of a certain currency may not be unique
10) Time, e.g. durable serves for production more than the period of time but it’s not included.
General price level is the weighted average of price of all goods and services in the economy. The price index at base period is set at 100.
Consumer price index (CPI) is the price level of consumer goods and services generally purchased by domestic households in a specified period. In HK, CPI(A), (B) and (C) reflects the price level for lower, medium and higher income group, covering 50%, 30% and 10% of household respectively. Composite CPI reflects the price level of products generally purchased by all the above households.
Implicit price deflator of GDP shows the price level of goods and services related to GDP in a specified period.
GDP deflator of the current period = Nominal GDP / Real GDP * 100
CPI VS GDP deflator
- CPI covers consumer goods while GDP covers all goods and services related to GDP
- CPI has a more fixed weights (in HK, the weights are adjusted every 5 years) while GDP has variable weights, depends on the composition of GDP.
- CPI is a better indicator of living standard since the cost of living is related to those consumer goods and services.
Growth rate refers to (new-old)/old * 100%, while growth rate in GDP is approx. equal to growth rate in population * growth rate in per capita GDP.
Saturday, 25 December 2010
Chemistry : Introduction to carbon chemistry
Carbon chemistry
Uniqueness of carbon:
1) Ability to catenate into a long chain, i.e., forming stable bonds with itself steadily (C-C bond), Note that Si can’t catenate since Si-Si bond is weak.
2) Forming multiple bonds since it can process sp^{3}, sp^{2} and sp hybridization
3) Forming 4 covalent bonds.
Nomenclature and expression:
- Condensed structural formula: omit C-C and C-H bond in the parent chain. E.g. CH_{3}CH_{3}.
Use bracket for branched chain, e.g. CH_{3}CH(CH_{3})CH_{3}
- Skeletal formula: omitting C atom, C-C bond and C-H bond.
Functional group is one or a group of toms that determines most properties of the compound.
A homologous series is a series of compounds that have the same functional group; each member differs from the next one by a –CH2- unit. For example, alkane: CH_{4}, CH_{3}CH_{3}…
They have the same general formula, shows gradual change in physical properties and similar in chemical properties.
Gradual change in physical properties down the homologous series include: darker in colour, more viscous, less volatile, less flammable and burns with a more sootier flame.
There’re two types of hydrocarbons, aliphatic hydrocarbon is those who doesn’t contain a benzene ring while aromatic hydrocarbon has at least one benzene ring. Hydrocarbons with π-bond are called unsaturated hydrocarbons.
Functional group: (R represents H or alkyl group, X represents halogen atom)
1) Alkane is not a functional group, has a general formula of CnH2n+2 (RH).
2) Alkene has general formula C_{n}H_{2n} (R_{2}C=CR_{2}), it contains double bond(s).
3) Alkyne has general formula C_{n}H_{2n-2} (RC≡CR), it contains triple bond(s)
4) Haloalkane has general formula C_{n}H_{2n+1}X (RX), it contains halogen atom(s).
5) Alkanol has general formula C_{n}H_{2n+1}OH (RO) it contains –OH group(s).
6) Alkanoic acid has general formula C_{n-1}H_{2n+1}COOH (RCOOH)
7) Aromatic hydrocarbon contains a benzene ring or phenyl (C_{6}H_{5}-) group.
Basic nomenclature: prefix – parent – unsaturated form - suffix
1) Parent chain is the longest chain that contains the functional group, triple and double bond. From 1 to 8 carbon: meth-, eth-, prop-, but-, pent-, hex-, hept- and oct-. For example, C8H18 is called octane.
2) Assigning number to the carbon atom on the carbon chain, functional group and π-bond should be assigned a number which is the smallest possible. For example, CH_{3}CH_{2}OH, C is assigned 1 but not 2.
3) Suffix: the functional group; for example, CH_{3}CH_{2}COOH has suffix –oic acid. When the functional group does not has a unique position, numbers are assigned before suffix. For example there’re two butene, namely but-2-ene (CH_{3}CH=CHCH_{3}) and but-1-ene (CH_{2}=CHCH_{2}CH_{3}), the double bond appears in the first and second C atom respectively.
4) Prefix: the substitutes, mainly branched chains, like CH_{3}CHClCH is called 2-chloropropane, since it has 3 C atom on parent chain (prop-), has no functional group (-ane), and the Cl atom is on the second C atoms. When there’re more than one substitutes,, we use alphabetical order to arrange them.
5) The unsaturated form shows the existence of π-bonds. Double bond is shown by –en-, it there’re more than one double bonds, we use di-, tri- and tetra-, e.g. if there’re three double bond and 2 triple bond, then the unsaturated form is –triendiyne.
6) The name of functional group uses –yl, like CH_{3}- is methyl and C_{2}H_{5}- is ethyl-
7) When there’re more than 1 function groups, -oic acid takes higher priority than –OH, then the –OH group will be the prefix.
8) For cycloalkanes, the functional group is assigned as the first C atoms.
More examples:
- CH_{3}C(CH_{3})_{2}CH_{2}CH_{3} has 4 C atom on parent chain, no functional groups, and 2 methyl group on the second C atom, so it’s called 2,2-dimethylbutane
- CH_{3}CH=CHCl has 3 C atoms on parent chain, belongs to alkene, and has 1 chlorine atom on the first C atom, so it’s called 1-chloroprop-1-ene.
- HOCH_{2}CH_{2}COOH is called 3-hydroxypropanoic acid.
- CH_{3}C(CH_{3})=CBrCOOH is called 2-bromo-3methylbut-2-enoic acid.
- The figure shown is 1-methylhexene, since the C atom connected to methyl group and double bond is assigned as the first carbon atom.
- Benzene is cyclohex-1,3,5-triene.
Reactions of alkane:
1) Combustion – C_{n}H_{2n+2}+(3n+1)/2 O_{2}→nCO_{2}+(n+1)H_{2}O, upon incomplete combustion, CO and C (unconsumed hydrocarbon) is produced, and sooty flame is observed. Combustion of alkane has made alkane into the main use of fuel.
2) Substitution (Halogenation):
- Initiation: Under UV light, X_{2 }→ 2X∙, X∙ refers to halogen free radicals.
- Chain reaction (propagation): RH (alkane) + X∙ → R∙ + HX and R + X_{2} → RX + X∙
- Termination occurs when two free radicals collides and combine: X∙+ X∙→X_{2}, X∙+R→RX or R∙+R’∙→R_{2}
- Under excess alkane, the major product is monosubstituted haloalkane and if halogen is in excess, haloalkanes with all H being substituted by halogen atoms will be the main product. For example, in excess Cl_{2}, CH_{4} will eventually change into CCl_{4}. If there’re excess CH_{4}, then major product will be CH_{3}Cl.
3) Cracking: Under heating of catalyst, when long-chain hydrocarbons pass through the catalyst, it cracks into shorter alkane, alkene and alkyne. It’s called catalytic cracking. It has the significant use of producing alkenes and extra fuels.
Reactions of alkene
1) Addition : When X_{2} is react with alkene in organic solvent, halogen atom are attached to the two C atom in the double bond. e.g.
It serves as test for unsaturation: Br_{2} in organic solvent. At the same time, alkenes under Pt/Pd/Rh/Ni as catalyst react with H_{2} to give alkanes.
2) Oxidation: CH_{2}=CH_{2} + [O] + H_{2}O → HOCH_{2}CH_{2}OH, the alkene is oxidized into a diol. It serves as a test for unsaturation: purple cold, diluted KMnO_{4}/H^{+} become colourless when added to unsaturated hydrocarbon. Other oxidizing agents include KMnO_{4}/OH^{-} and OsO_{4}.
Applications – fossil fuels
Coal is formed by dead land animals in a million years ago , covered by mud and under heat, pressure and effects of bacteria. Petroleum and natural gas is formed in the similar way, but it requires sea dead animals instead of that from land.
Uses of different fraction of petroleum
1) Refinery gas – fuel (LPG) 2) Petrol – Car fuel
3) Naphtha – raw material of town gas 4) Kerosene – domestic fuel, aircraft fuel
5) Diesel oil – fuel for heavy vehicles 6) Fuel oil – fuel for ships and power stations
7) Wax, lubricating oils – candles 8) Bitumen – Road construction, roofs
Exothermic reaction is the reaction that heat is released while heat is taken in during endothermic reaction.
Air pollutants
1) CO and C from incomplete combustion in cars, CO is toxic.
2) Unconsumed hydrocarbons from cars like benzene are carcinogenic.
3) Respirable suspended particle (RSP) reduces visibility and damage respiratory system.
4) Oxides of nitrogen (NO_{x}) from car engines, gives acid rain (2NO_{2}+H_{2}O→HNO_{2}+HNO_{3}), irritate respiratory system and forms photochemical smog with unconsumed hydrocarbon.
5) SO_{2} from combustion of impure fuel, is irritating and form acid rain (SO_{2}+H_{2}O→H_{2}SO_{3})
6) Heavy metal oxides (mainly PbO) from impurities: toxic
Solutions:
1) Catalytic convertors (Pt/Pd/Rh) engages complete combustion and convert CO, NO_{x} into less harmful CO_{2} and N_{2}.
2) Scrubbers or desulphurization systems in the power stations: SO_{2} is neutralized by CaO.
3) Electrostatic precipitators in power stations and particulate removal devices in motor vehicles remove particulates.
4) Using lead-free, low-sulphur content fuels for cards, low nitrogen oxide burners in power stations.
I've got some hard time to select those elementary reactions since it's hard to exclude some reaction when you're taking "addition" or other kinds of reactions. For example, KMnO4/H+ further oxidizes diols, so KMnO4/OH- is a better test for unsaturation, but it's not on the book.
Thursday, 23 December 2010
Economics: National Income statistic I
Macroeconomics – study on aggregate output and general price level
Stock and flow as measures of the economic variables: stocks is measured at an instant of time but flow is measured in a period of time. For example, wealth is a stock but income is a flow (measured “per month”, measuring income at a point of time is meaningless)
GDP and GNP as the measurement of aggregate output:
1) Gross domestic product is the total market value of production of all residential producing units (RPU) of a country of territory in a specified period (usually 1 year). RPU refers to individuals or organizations with engaged to product and maintain the economic interest at the territory. Note that GDP counts RPU as units, not individuals.
2) Gross national product is the total market value of production of all residents of a country of territory in a specified period. Residents refers to individuals which maintains the economic interest at the territory, irrespective to their nationality.
Stuffs that excluded in GDP:
1) Past inventories (out of the period) are not counted.
2) Second-hard goods since the production value is counted in its first transaction, and this prevents double counting.
3) Unpaid household services for self-consumption
4) Intermediate products since they’re not on sale on product market (only sold on factor market), and this prevents double counting.
5) Financial assets, capital gain and transfer payments, since they don’t involve production.
Approaches to measure GDP
1) Expenditure approach: C+I+G+(X-M) = C+I+G+NX
C: private consumption expenditure (on households and private non-profit institutions). Note that renting premises and owner-occupied premises are both counted in C.
I: Gross investment expenditure refers to:
- Gross fixed investment expenditure includes fixed capital like (all) building, construction, machinery, equipment, software, as well as transfer of ownership of land and buildings like stamp duties.
- Change in inventories
It can also be refers to Depreciation + Net investment expenditure.
G: government consumption expenditure refers to expenditure on consumption by government departments that do not engage in market activities (e.g., Fire Services Dept.) and non-profit quasi-government institutions (e.g., Hospital Authority). Expenditure of other government dept. will be counted as I.
X: Exports, include domestic export of goods, export of services and re-export of goods.
M: Imports, includes imports of services and goods. NX: Net exports
2) Production approach: sum of value-added of all RPU
Value-added = Value of output – value of intermediate consumption
e.g. A sales imports a bottle of wine at $60 and sold It at $80, then his value-added is $20.
3) Income approach: Rent + compensation of employees + capital consumption allowance + interest + profit (before tax)
Theoretically value obtained from the three approaches are the same.
Market price VS Factor cost
Since market price = factor cost + tax – subsidies, say P = C + t – s, then ΣPQ = ΣCQ + Σt – Σs, therefore GDP at factor cost = GDP at market price (abbreviated as GDP) – tax + subsidies.
GDP VS GNP
GNP = GDP + factor income earned by residents outside the economic territory (factor income from aboard) – factor income earned by non-residents within the economic territory (factor income paid aboard) = GDP + Net income from aboard (NIFA)
Monday, 20 December 2010
Notes on chem notes
I would like to follow the following sequences for my chemistry notes:
1) Nomenclature and properties
2) Reaction
3) Mechanism
4) Reigonochemistry
5) Applications
This follows the sequences of AL-chem topics, but I think the mechanism is greatly simplified since the concept of electrophile and nucleaophile is out syb.
For example unit 25-27 will be arranged like this
1) Homologus series
2) Nomenclature of alkane, alkene, alcohol carboxylic acid
3) Reaction of alkane
4) Reaction of alkene
5) Application : Fossil fuels
Also I want to include some extra reactions that would be useful:
e.g. the reaction that changes alkene to diols:
RCH=CHR + [O] + H2O -> R(CH(OH))2R + other product
Other than KMnO4/H+, OsO4 is also an alternative choice.
1) Nomenclature and properties
2) Reaction
3) Mechanism
4) Reigonochemistry
5) Applications
This follows the sequences of AL-chem topics, but I think the mechanism is greatly simplified since the concept of electrophile and nucleaophile is out syb.
For example unit 25-27 will be arranged like this
1) Homologus series
2) Nomenclature of alkane, alkene, alcohol carboxylic acid
3) Reaction of alkane
4) Reaction of alkene
5) Application : Fossil fuels
Also I want to include some extra reactions that would be useful:
e.g. the reaction that changes alkene to diols:
RCH=CHR + [O] + H2O -> R(CH(OH))2R + other product
Other than KMnO4/H+, OsO4 is also an alternative choice.
Saturday, 18 December 2010
18-12-10
同個fd玩鬥估賽果，如果我輸了的話要寫一篇TCG導論……
TCG (tatical card game) 的致勝之道不在於你有多強的卡牌，而是在於你卡組能做出必殺的機率高低，以及最差情況下的計劃。不要被電視蒙騙，沒有人永遠能在最後時刻抽出王牌反勝，但將這機會放到最大，你就是勝出的人。
分析TCG相信比RPG更為困難，因為能力很大程度上為運氣所主宰，因此在我看來卡牌遊戲的首要目的可以理解成「對抗命運」：當你一直抽爛牌時，你又會如何處理呢？
……
TCG (tatical card game) 的致勝之道不在於你有多強的卡牌，而是在於你卡組能做出必殺的機率高低，以及最差情況下的計劃。不要被電視蒙騙，沒有人永遠能在最後時刻抽出王牌反勝，但將這機會放到最大，你就是勝出的人。
分析TCG相信比RPG更為困難，因為能力很大程度上為運氣所主宰，因此在我看來卡牌遊戲的首要目的可以理解成「對抗命運」：當你一直抽爛牌時，你又會如何處理呢？
……
Tuesday, 14 December 2010
Multiple geometric summation
Recently I found that applications of geometric summation has been a popular topic among senior interschool competitions, the idea to solve them is actually easy, but only a few students are able to tackle them.
1)Recall: geometric summation
We mainly concerns infinite sum, so for all |r|<1, a+ar+ar^2+... = a(1+r+r^2+...) = a/(1-r)
Example: 1+1/2+1/4+... = 2.
2)A special case
We should note that 1/r+1/r^2+1/r^3+... = 1/(r-1).
Example: 1/8+1/64+1/512+... = 1/7.
3)Summation we uses the sigma sign to denote the summation.
e.g. Σ k (from 0 to 3) = 0+1+2+3 = 6
Double summation: Σ(x: from a to b)Σ(y: from c to d) f(x,y) = f(a,c)+f(a,c+1)+...+f(a+d)+f(a+1,c)+...+f(b,d)
Sometimes the inner summation's first and last term is bonded by the outer summation.
e.g. Σ (x: from a to c) Σ (y: from x to 2x) f(x,y) = f(a,a)+f(a,a+1)+...+f(a,2a)+f(a+1,a+1)+...+f(c,2c).
The following geometric summation, unless specified, adds from 1 to infinity.
4)Classic doubles
e.g. Evaluate Σ i / 2^i.
You may find that it's not a strict G.S. but we can solve it by G.S. by spliting terms.
1/2 + 2/4 + 3/8 + 4/16
= (1/2+1/4+1/8+1/16+...)+(1/4+1/8+1/16+...)+(1/8+1/16+...)+...
= (1)(1/2+1/4+1/8+1/16+...)+(1/2)(1/2+1/4+1/8+...)+(1/4)(1/2+1/4+1/8+...)+...
=(1+1/2+1/4+...)(1/2+1/4+1/8+...)
=(1)(2)=2
You may notice that G.S. turn sum to one terms, so double G.S. turns the sum into product of two terms.
When it's in forms of Σi/10^k, the result is k/(k-1)^2, which suits "product of two terms" and the lemma of special case on G.S..
5)Double summation in terms of summation
If we want to tackle double G.S. by summation we will have to use double summation.
e.g. Evaluate Σ i/3^i.
Σ i/3^i
=Σ 1/3^i + 1/3^i +... (i terms)
=Σ (i:: from 1 to infinity) Σ (j: from 1 to i) 1/3^j
=Σ (i: from 1 to infinity) Σ (j from i to infinity) 1/3^j (Note: this can be shown on a grpah.)
=Σ 1/(3-1)3^(i-1)
=(1/2) Σ 1/3^(i-1)
=(1/2)(3/(3-1))
=3/4
6)Further applications:
e.g. Evaluate 7/10^2 + 77/10^4 + 777/10^6+...
7/10^2 + 77/10^4 + 777/10^6 +...
=(7/10^2+7/10^4+7/10^6+...) + (70/10^4+70/10^6+...) + (700/10^6+700+10^8+...)+...
=7(1/10^2+1/10^4+...)+7/10(1/10^2+1/10^4+...)+7/100(1/10^2+1/10^4+...)=...
=7(1+1/10+1/100+...)(1/10^2+1/10^4+...)
=7(11/10)(1/99)
=7/90
7)Extension: multiple G.S.
e.g. Evaluate Σi^2 / 2^i.
Does it converge? Yes since (i^2/2^i)' is eventually negative and tends to 0.
Use this identity: i^2 = 1+3+5+...+(2i-1).
1/2+4/4+9/8+16/16+...
=1/2+1/4+3/4+1/8+3/8+5/8+1/16+3/16+5/16+7/16+...
=(1/2+1/4+1/8+...)+3(1/4+1/8+...)+5(1/8+1/6)+...
=1+3/2+5/4+7/8+9/16+11/25+...
=(1+1/2+1/4+1/8+...)+2(1/2+1/4+1/8+...)+2(1/4+1/8+...)+...
=2+2+1+1/2+1/4+...=6
Note that three times of G.S. is used here.
8)More to explore
-Prove the special case of G.S.
-Evaluate Σ (n^2+n)/7^(n+1).
-Evaluate Σ i^3/3^i and Σ i^4/4^i.
-Find the rage of y'<0 of the graph y = i^x/x^i, where x is a positive real. Hence or otherwise prove Σ (x : from 2 to fintiniy) Σ i^x/x^i converges.
-Show that (x-1)^xΣ i^x/x^i is an integer in terms of i) "product of n terms" and "calculation analysis"
-Explore more about polylogarithm function and this G.S.
1)Recall: geometric summation
We mainly concerns infinite sum, so for all |r|<1, a+ar+ar^2+... = a(1+r+r^2+...) = a/(1-r)
Example: 1+1/2+1/4+... = 2.
2)A special case
We should note that 1/r+1/r^2+1/r^3+... = 1/(r-1).
Example: 1/8+1/64+1/512+... = 1/7.
3)Summation we uses the sigma sign to denote the summation.
e.g. Σ k (from 0 to 3) = 0+1+2+3 = 6
Double summation: Σ(x: from a to b)Σ(y: from c to d) f(x,y) = f(a,c)+f(a,c+1)+...+f(a+d)+f(a+1,c)+...+f(b,d)
Sometimes the inner summation's first and last term is bonded by the outer summation.
e.g. Σ (x: from a to c) Σ (y: from x to 2x) f(x,y) = f(a,a)+f(a,a+1)+...+f(a,2a)+f(a+1,a+1)+...+f(c,2c).
The following geometric summation, unless specified, adds from 1 to infinity.
4)Classic doubles
e.g. Evaluate Σ i / 2^i.
You may find that it's not a strict G.S. but we can solve it by G.S. by spliting terms.
1/2 + 2/4 + 3/8 + 4/16
= (1/2+1/4+1/8+1/16+...)+(1/4+1/8+1/16+...)+(1/8+1/16+...)+...
= (1)(1/2+1/4+1/8+1/16+...)+(1/2)(1/2+1/4+1/8+...)+(1/4)(1/2+1/4+1/8+...)+...
=(1+1/2+1/4+...)(1/2+1/4+1/8+...)
=(1)(2)=2
You may notice that G.S. turn sum to one terms, so double G.S. turns the sum into product of two terms.
When it's in forms of Σi/10^k, the result is k/(k-1)^2, which suits "product of two terms" and the lemma of special case on G.S..
5)Double summation in terms of summation
If we want to tackle double G.S. by summation we will have to use double summation.
e.g. Evaluate Σ i/3^i.
Σ i/3^i
=Σ 1/3^i + 1/3^i +... (i terms)
=Σ (i:: from 1 to infinity) Σ (j: from 1 to i) 1/3^j
=Σ (i: from 1 to infinity) Σ (j from i to infinity) 1/3^j (Note: this can be shown on a grpah.)
=Σ 1/(3-1)3^(i-1)
=(1/2) Σ 1/3^(i-1)
=(1/2)(3/(3-1))
=3/4
6)Further applications:
e.g. Evaluate 7/10^2 + 77/10^4 + 777/10^6+...
7/10^2 + 77/10^4 + 777/10^6 +...
=(7/10^2+7/10^4+7/10^6+...) + (70/10^4+70/10^6+...) + (700/10^6+700+10^8+...)+...
=7(1/10^2+1/10^4+...)+7/10(1/10^2+1/10^4+...)+7/100(1/10^2+1/10^4+...)=...
=7(1+1/10+1/100+...)(1/10^2+1/10^4+...)
=7(11/10)(1/99)
=7/90
7)Extension: multiple G.S.
e.g. Evaluate Σi^2 / 2^i.
Does it converge? Yes since (i^2/2^i)' is eventually negative and tends to 0.
Use this identity: i^2 = 1+3+5+...+(2i-1).
1/2+4/4+9/8+16/16+...
=1/2+1/4+3/4+1/8+3/8+5/8+1/16+3/16+5/16+7/16+...
=(1/2+1/4+1/8+...)+3(1/4+1/8+...)+5(1/8+1/6)+...
=1+3/2+5/4+7/8+9/16+11/25+...
=(1+1/2+1/4+1/8+...)+2(1/2+1/4+1/8+...)+2(1/4+1/8+...)+...
=2+2+1+1/2+1/4+...=6
Note that three times of G.S. is used here.
8)More to explore
-Prove the special case of G.S.
-Evaluate Σ (n^2+n)/7^(n+1).
-Evaluate Σ i^3/3^i and Σ i^4/4^i.
-Find the rage of y'<0 of the graph y = i^x/x^i, where x is a positive real. Hence or otherwise prove Σ (x : from 2 to fintiniy) Σ i^x/x^i converges.
-Show that (x-1)^xΣ i^x/x^i is an integer in terms of i) "product of n terms" and "calculation analysis"
-Explore more about polylogarithm function and this G.S.
fin.
Subscribe to:
Posts (Atom)