Question

Find the interval of convergence of P∞
n=0
x
n
n+4 .

Answer 1

Given series is $\sum^\infin_{n=0} \frac{x^n}{n+4}.$

Let $a_n = \frac{1}{n+4}$

$\implies a_{n+1} = \frac{1}{n+5}$

So, Radius of convergence is $R= \lim_{n \to \infin} \frac{a_n}{a_{n+1}} = \lim_{n \to \infin} \frac{n+5}{n+4} = \lim_{n \to \infin} \frac{1+5/n}{1+4/n} =1$ .

So, series is convergence on $(-1,1)$ is convergent and on $(-\infin,-1) \cup(1,\infin)$ series is divergent.

Now, at $x=1$ series becomes $\sum^\infin_{n=0} \frac{1}{n+4}$ which is divergent by p-test.

At $x=-1$ series becomes $\sum^\infin_{n=0} \frac{(-1)^n}{n+4}$, which is convergent by Leibniz's test because $<\frac{1}{n+4}>$ is decreasing sequence and$\frac{1}{n+4} \to 0$ as $n\to \infin$.

So interval of convergence is $[-1,1)$ .