# Find the radius and interval of convergence of the summation \sum_{n=1}^\infty \frac{...

## Question:

Find the radius and interval of convergence of the summation {eq}\sum_{n=1}^\infty \frac{ (x+2)^n}{n4^n} {/eq}

## The Radius of Convergence for a Power Series:

Interval of convergence for a power series is a set, where all values of the set covergence the power series. In this problem, we will use the Ratio test to find the interval of convergence. Which states that, If there exists an N so that for all {eq}n\geq N, a_{n}\neq 0 {/eq}

{eq}\lim _{n\rightarrow \infty }\left| \dfrac {a_{n+1}}{a_{1}}\right| =L {/eq}

If {eq}L<1 {/eq} then {eq}\sum a_{n} {/eq} coverges,

If {eq}L>1 {/eq} then {eq}\sum a_{n} {/eq} diverges,

If {eq}L=1 {/eq} then the test is inconclusive.

We are given;{eq}\sum_{n=1}^{\infty}\frac{(x+2)^n}{n 4^n} {/eq}

Use the Series Ratio test;

{eq}\displaystyle a_{n}= \frac{(x+2)^n}{n 4^n} {/eq}

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