Sum of x raised to odd powers till infinity where x is less than 0

Question

I have an geometric series which has the following form

0.5^3 + 0.5^5 + 0.5^7 + 0.5^9 .... 0.5^inf

Is there a formula to find the sum of this series ?

Answer 1

It is a simple geometric progression $\{a_n\}$, its general term is given by $a_n=a_1q^{n-1}$ , correspondingly, summation formula is $S_n=a_1\frac{1-q^n}{1-q}$. Specifically, when $\vert q\vert<1$, take limitation on both sides, one obtains $S_{\infty}=\frac{a_1}{1-q}$.

So when it comes to your problem, $a_1=0.5^3, q=0.5^2, S_{n}=0.5^3\times\frac{1-0.5^{2n}}{1-0.5^2}, S_{\infty}=\frac{0.5^3}{1-0.5^2}=\frac{1}{6}$ .