Question

Find all values of x for which the series converges, and find the sum of the series for those values of x.

Answer 1

1/(x^2)+4/(x^3)+16/(x^4)+64/(x^5)+256/(x^6)

Answer 2

converges for x < ? or x> ?

Answer 3

what is the original series

Answer 4

a series has infinite terms, does this series go on

Answer 5

yes :)

Answer 6

ok it looks like your series is
$$
\Large{\sum_{n=0}^{\infty} \frac{4^{n}}{x^{n+2}} }
$$

Answer 7

to find where it converges , we can use ratio test

Answer 8

alternatively, you can rewrite the series

Answer 9

ok it looks like your series is
$$
\Large{\sum_{n=0}^{\infty} \frac{4^{n}}{x^{n+2}}
=
\sum_{n=0}^{\infty} \frac{4^{n}}{x^{n}\cdot x^2}= \sum_{n=0}^{\infty} (\frac{4}{x})^n\cdot \frac{1}{x^2} \\
= }
$$

Answer 10

i got it ! thank you :)