Question

I'm supposed to find the radius of convergence and radius of convergence.
\[\sum_{n=0}^{\infty} \frac{(-1)^{n}x^{n}}{n+1}\]

Answer 1

I started with the ratio test

Answer 2

this is what I came up with \[\lim_{n \rightarrow \infty} \left|x\right| \frac{n+1}{n+2}\]

Answer 3

=x

Answer 4

right? \[\lim_{n \rightarrow \infty} \left|x\right| \frac{n/n+1/n}{n/n+2/n}\]

Answer 5

Seems right.

Answer 6

where from here?

Answer 7

The ratio test states that, if we call the limit L, so L=|x| here,

if L<1 the series converges

if L>1 the series diverges

if L=1 inconclusive

Answer 8

so as long as x<1

Answer 9

To be precise: |x|<1, so -1<x<1

Answer 10

just curious why -1<x ?

Answer 11

|x|<1 is equivalent to -1<x<1

if x=-2 |x|=|-2|=2 is not smaller than 1.

Answer 12

ok

Answer 13

You can see that the series diverges for a large negative x, -10 for example: 5+100/3+1000/4+...

Answer 14

yes

Answer 15

Also the series might converge for x=1 or -1 as well, the test is inconclusive there.

Answer 16

something about harmonic series and alternating harmonic series?

Answer 17

yes