Determine the radius and interval of convergence for each of the following power series. sum_{n=0}^inftyfrac{2^n(x-3)^n}{sqrt{n+3}}

Mylo O'Moore

Mylo O'Moore

Answered question

2021-03-08

Determine the radius and interval of convergence for each of the following power series.
n=02n(x3)nn+3

Answer & Explanation

AGRFTr

AGRFTr

Skilled2021-03-09Added 95 answers

Given:
The power series is:
n=0an=n=02n(x3)nn+3
For the given power series, we must ascertain the radius and interval of convergence.
We know that,
When
limn|an+1an|<1
Once the power series converges, we must determine the interval for x such that it does.
We have,
an=2n(x3)nn+3 and an+1=2(n+1)(x3)(n+1)(n+1)+3
Now,
limn|an+1an|<1
limn|2(n+1)(x3)(n+1)(n+1)+32n(x3)nn+3|<1
limn|2(x3)n+3n+4|<1
limn|2(x3)1+3n1+4n|<1
|2(x3)|<1
1<2(x3)<1
12
12+3
52
The interval for the radius for convergence is (52,72)
Now consider,
For x=3 the radius of convergence is:
n=0an=n=02n(x3)nn+3
We get, R is the radius of the convergence which is equal to zero
Hence, the radius of the convergence is zero for x=3 and the interval for the convergence is
(52,72)

Jeffrey Jordon

Jeffrey Jordon

Expert2021-12-27Added 2605 answers

Answer is given below (on video)

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