Find a sum of fractional series

Question

I have a series:
$$sum_{n=1}^{infty} frac{1}{4^{2n-1}}$$
I know that $sum_{n=1}^{infty} frac{1}{4^{n}} = frac{1}{1-frac{1}{4}}=frac{4}{3}$, but what should I use in my case?

Toby Mak · Answer

If in doubt, you can always write out the first few terms of the series:
$$S = frac{1}{4} + frac{1}{4^3} + frac{1}{4^5} + frac{1}{4^7} cdots tag{1}$$
$$frac{1}{4^2}S = frac{1}{4^3} + frac{1}{4^5} + frac{1}{4^7}cdots tag{2}$$
Since the series converges as $|r| < 1$, $(1) - (2)$ gives:
$$frac{15}{16}S = frac{1}{4} Rightarrow S = frac{1}{4} cdot frac{16}{15} = frac{4}{15}$$

Find a sum of fractional series

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