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Question Number 109855 by aurpeyz last updated on 26/Aug/20

Answered by aurpeyz last updated on 26/Aug/20

$$Cananyoneplsexplainthis?itispertaining$$$$binomialexpansionfornegativepowers$$

Commented by mathdave last updated on 26/Aug/20

$$eitherforpositiveornegativedegreepower$$$$alwaysusethisgeneralizationformular$$$${(x+y)}^n=\frac{x^n}{0!}+\frac{{nx}^{n-1}y}{1!}+\frac{n(n-1)x^{n-2}{y}^2}{2!}+\frac{n(n-1)(n-2)x^{n-3}{y}^3}{3!}..........$$

Answered by Her_Majesty last updated on 26/Aug/20

$$\underset{j=0}{\sum ^n}{x}^j=\frac{x^{n+1}-1}{x-1}$$$${lim}_{n\to \mathrm{\infty }}\frac{x^{n+1}-1}{x-1}onlyexistsif\mid x\mid <1$$$$inthiscase{x}^{n+1}=0\Rightarrow \underset{j=0}{\sum ^{\mathrm{\infty }}}{x}^j=\frac{1}{1-x}with\mid x\mid <1$$

Answered by Rio Michael last updated on 26/Aug/20

$$\mathrm{Given}\mathrm{the}\mathrm{general}\mathrm{binomial}\mathrm{expansion}{(1+x)}^n,\mathrm{then}\mathrm{general}$$$$\mathrm{term}\mathrm{expressed}\mathrm{as}\mathrm{a}\mathrm{summation}\mathrm{is}\mathrm{given}\mathrm{as}$$$${(1+x)}^n=\underset{k=0}{\sum ^n}{}^n{C}_k{x}^{n-k}$$$$\mathrm{It}\mathrm{is}\mathrm{known}\mathrm{that}\mathrm{the}\mathrm{series}\mathrm{is}\mathrm{convergent}\mathrm{if}\mid x\mid <1.$$$$\mathrm{If}\mathrm{we}\mathrm{wish}\mathrm{to}\mathrm{expand}\mathrm{such}\mathrm{that}\mid x\mid >1,\mathrm{then}\mathrm{it}$$$$\mathrm{must}\mathrm{be}\mathrm{expanded}\mathrm{in}\mathrm{terms}\mathrm{of}\frac{1}{x},\mathrm{this}\mathrm{works}\mathrm{since}\frac{1}{x}<1$$$$\mathrm{and}\mathrm{will}\mathrm{be}\mathrm{carried}\mathrm{out}\mathrm{for}\mathrm{the}{\mathrm{expansion}}^{\prime }\mathrm{s}\mathrm{of}{(1+x)}^{-n}\mathrm{only}.$$

Commented by aurpeyz last updated on 26/Aug/20

$$pls.\mathrm{What}\mathrm{is}\mathrm{the}\mathrm{implication}\mathrm{of}\mathrm{this}\mathrm{statement}?$$$$\mathrm{Does}\mathrm{that}\mathrm{mean}\mathrm{that}\mathrm{i}\mathrm{have}\mathrm{to}\mathrm{factorize}$$$$\mathrm{any}\mathrm{question}\mathrm{on}\mathrm{negative}\mathrm{power}\mathrm{binomial}\mathrm{to}$$$$\mathrm{get}\mathrm{the}\mathrm{form}\frac{1}{\mathrm{x}}\mathrm{in}\mathrm{other}\mathrm{to}\mathrm{get}\mathrm{the}\mathrm{right}\mathrm{ans}$$

Commented by aurpeyz last updated on 26/Aug/20

$$\mathrm{That}\mathrm{is}\mathrm{exactly}\mathrm{where}\mathrm{I}\mathrm{am}\mathrm{getting}\mathrm{confused}.$$$$\mathrm{Anytime}\mathrm{i}\mathrm{have}\mathrm{a}\mathrm{negative}\mathrm{power}.\mathrm{I}\mathrm{always}$$$$\mathrm{factorize}\mathrm{the}\mathrm{binomial}\mathrm{to}\mathrm{get}\mathrm{the}\mathrm{form}\frac{1}{\mathrm{x}}\mathrm{in}$$$$\mathrm{expansion}.\mathrm{but}\mathrm{the}\mathrm{problem}\mathrm{is}\mathrm{that}\mathrm{it}\mathrm{doesnt}\mathrm{work}$$$$\mathrm{for}\mathrm{all}\mathrm{kinds}\mathrm{of}\mathrm{problem}$$

Commented by Rio Michael last updated on 26/Aug/20

$$\mathrm{No}\mathrm{its}\mathrm{expansion}\mathrm{as}\mathrm{you}\mathrm{would}\mathrm{for}\mathrm{anyother}\mathrm{problem}.$$$$\mathrm{Most}\mathrm{at}\mathrm{times}\mathrm{it}\mathrm{will}\mathrm{appear}\mathrm{in}\mathrm{terms}\mathrm{of}\frac{1}{x},\mathrm{if}\mathrm{it}{\mathrm{doesn}}^{\prime }\mathrm{t}$$$$\mathrm{then}\mathrm{either}\mathrm{the}\mathrm{series}{\mathrm{doesn}}^{\prime }\mathrm{t}\mathrm{converge}\mathrm{or}{\mathrm{it}}^{\prime }\mathrm{s}\mathrm{got}\mathrm{a}\mathrm{particular}$$$$\mathrm{range}\mathrm{of}\mathrm{numbers}\mathrm{for}\mathrm{which}\mathrm{it}\mathrm{converges}.$$

Commented by aurpeyz last updated on 27/Aug/20

$$Sir.PlscheckquestionQ.110109helpmewiththeexpansion$$

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