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This question concerns the infinite geometric series formula. It turns out there is a nice formula for the sum of an infinite geometric series.

Consider the infinite geometric series $1+r+r^2+r^3+\cdots$ For this series to converge, what must be the true about $r$? Explain.

I have totally no clue what it means. Can anyone help me with this question? Thank you very much.

dfeuer
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user108186
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  • That series (and any geometric series) converges iff $;|r|<1;$ (the absolute value of the ratio is less than one). – DonAntonio Nov 12 '13 at 05:00
  • one part of this question ask me to find the closed-form for this question is (1-r^n+1)/(1-r). Then it ask you to find the value to which this infinite geometric series converges by using the formula to allowing n to approach infinity. English is my second languages so it is really hard to understand the questions so may you guys help me? – user108186 Nov 12 '13 at 05:05
  • so |r| will smaller than 1 and bigger than zero? @ DonAntonio – user108186 Nov 12 '13 at 05:07

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Ok, so

$$\sum_{k=0}^n r^k=\frac{1-r^n}{1-r}\;,\;\;r\neq 1$$

and this is true always. Now

$$\lim_{n\to\infty}\frac{1-r^n}{1-r}=\begin{cases}\frac1{1-r}&,\;\;|r|<1\\{}\\\pm\infty&,\;\;|r|>1\end{cases}$$

and from the above is clear the infinite geometric series converges iff $\;|r|<1]\;$ ,as already mentioned.

DonAntonio
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  • one part of this question ask me to find the closed-form for this question is (1-r^n+1)/(1-r). Then it ask you to find the value to which this infinite geometric series converges by using the formula to allowing n to approach infinity. English is my second languages so it is really hard to understand the questions so may you help me with this one too? Thank you so much. – user108186 Nov 12 '13 at 05:15
  • Prove the first formula above by induction, @user108186 : or in the form $$r^n-1=(r-1)(r^{n-1}+r^{n-2}+\ldots +r+1)$$ – DonAntonio Nov 12 '13 at 05:18