Question

\sum_{n=1}^{\infty } ((-1^n)/n)x^n Find the interval of convergence.

Answer 1

Answer:

(-1,1).

Step-by-step explanation:

We need to calculate $\lim_{n \to \infty}\frac{| a_{n+1}|}{| a_{n}|} = \frac{1}{R}$ where R is the radius of convergence.

$\lim_{n \to \infty}\frac{\frac{|(-1)^{n+1}|}{|n+1|}}{\frac{|(-1)^{n}|}{|n|}}$

$\lim_{n \to \infty}\frac{\frac{1}{|n+1|}}{\frac{1}{|n|}}$

$\lim_{n \to \infty}\frac{|n|}{|n+1|}$

Applying LHopital rule we obtaing that the limit is 1. So $1=\frac{1}{R}$ then R = 1.

As the serie is the form $(x+0)^{n}$ we center the interval in 0. So the interval is (0-1,0+1) = (-1,1). We don't include the extrem values -1 and 1 because in those values the serie diverges.