Question

Find the interval of convergence of the power series n=1∑[infinity]​(x+2)ⁿ/n4ⁿ

Answer 1

The interval of convergence of the power series is (-6, 2).

Step-by-step explanation:

The given series is:

∑[infinity] (x+2)ⁿ/n4ⁿ

To find the interval of convergence, we can use the ratio test:

|(x+2)ⁿ⁺¹/n⁺¹4ⁿ⁺¹| / |(x+2)ⁿ/n4ⁿ|

Taking the limit as n approaches infinity, we get:

|(x+2)| / 4

For the series to converge, the absolute value of this ratio must be less than 1. Therefore, we have:

|(x+2)| / 4 < 1

Simplifying this inequality, we find:

-4 < x+2 < 4

-6 < x < 2

So, the interval of convergence is (-6, 2).