TransWikia.com

Find a sum of fractional series

Mathematics Asked by manabou11 on November 6, 2021

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?

One 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:

$$frac{15}{16}S = frac{1}{4} Rightarrow S = frac{1}{4} cdot frac{16}{15} = frac{4}{15}$$

Answered by Toby Mak on November 6, 2021

Add your own answers!

Ask a Question

Get help from others!

© 2024 TransWikia.com. All rights reserved. Sites we Love: PCI Database, UKBizDB, Menu Kuliner, Sharing RPP