Find the interval of convergence of the series \sum_{n=0}^\infty \frac{(x+5)^n}{4^n}

Question:

Find the interval of convergence of the series {eq}\sum_{n=0}^\infty \frac{(x+5)^n}{4^n} {/eq}

Interval of Convergence for a Power Series:

In this problem, we will use the Root 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}

$$\lim _{n\rightarrow \infty }\left| {a_{n}}\right|^\dfrac{1}{n} =L $$

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.

Answer and Explanation:

We are given {eq}\displaystyle \sum_{n=0}^{\infty} \frac{(x+5)^n}{4^n} {/eq}

So

{eq}\displaystyle a_{n}= \frac{(x+5)^n}{4^n} {/eq}

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from PSAT Prep: Tutoring Solution

Chapter 10 / Lesson 13
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