Find the interval of convergence of the power series. sum_{n=1}^inftyfrac{(-1)^nx^n}{n}

Amari Flowers

Amari Flowers

Answered question

2020-10-27

Find the interval of convergence of the power series.
n=1(1)nxnn

Answer & Explanation

aprovard

aprovard

Skilled2020-10-28Added 94 answers

The given series is n=1(1)nxnn
Compare the above series with its standard form n=1(1)nanxn and obtain that an=1n
Obtain the radius of convergence as follows.
R=limnanan+1
=limn1n×n+11
=limn(1+1n)
=1+0
=1
Thus, the radius of convergence is 1.
That is, the series n=1(1)nxnn converges for all x in (-1,1).
Now check at its end points as follows.
at x=1, the series become n=1(1)nn which is convergent.
at x=-1, the series become n=1(1)2nn=n=11n which is divergent.
Thus, the interval of convergence is (−1, 1].

Jeffrey Jordon

Jeffrey Jordon

Expert2021-12-27Added 2605 answers

Answer is given below (on video)

Do you have a similar question?

Recalculate according to your conditions!

Ask your question.
Get an expert answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?