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so I have this problem. It's a Geometric Series:

$$x-x^{15}+x^{29}-x^{43}+x^{57}+\cdots$$

I can see that the exponent increases by 14 every time.

So I rewrote it like this:

$$x^1-x^1(x^1)^{14}+x^1(x^2)^{14}-x^1(x^3)^{14}+x^1(x^4)^{14}+\cdots$$

I figured that it converges for $|x|<1$

However, I'm not sure how to calculate the sum.

Thanks

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  • $\begingroup$ Note: $1^{14}=1$, $2^{14}=16384$. Perhaps you mean $1\cdot14$ and $2\cdot14$. I.e. $(x^1)^{14}=x^{1\cdot14}$ and $(x^2)^{14}=x^{2\cdot14}$. $\endgroup$
    – Peter Huxford
    Commented Nov 2, 2014 at 4:32
  • $\begingroup$ @Peter, wow sorry for that mistake. I will fix it right now $\endgroup$
    – KFC
    Commented Nov 2, 2014 at 4:33

2 Answers 2

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You can again rewrite it as $$\sum_{k=0}^{\infty} x(-x^{14})^{k}=\frac{x}{1+x^{14}}$$

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This is an infinite geometric series with first term $a=x$ and common ratio $r= - x^{14}$. So the series coverges iff |r|<1 i.e. $|-x^{14}|<1$ i.e. |x|<1. In that radius of convergence the sum of the series will be as follows: $$S= \frac{a}{1-r}$$ $$\Rightarrow S = \frac{x}{1-(-x^{14})}$$ $$\Rightarrow S= \frac{x}{1+x^{14}}$$

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