Geometric Series Sum

Question

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

Answer 1

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

Answer 2

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: $$S= \frac{a}{1-r}$$ $$\Rightarrow S = \frac{x}{1-(-x^{14})}$$ $$\Rightarrow S= \frac{x}{1+x^{14}}$$