Find the interval of convergence of the power series. sum_{n=0}^inftyfrac{(3x)^n}{(2n)!}

bobbie71G

bobbie71G

Answered question

2021-02-03

Find the interval of convergence of the power series.
n=0(3x)n(2n)!

Answer & Explanation

Laith Petty

Laith Petty

Skilled2021-02-04Added 103 answers

Interval of convergence using ratio test: If
limn|an+1an|=L
If L<1 then the series is absolutely convergent
If L>1 then the series is absolutely divergent
Given that
an=(3x)n(2n)!
So an+1=(3x)n+1(2n+2)!
Now evaluate |an+1an|, so
|an+1an|=|(3x)n+1(2n+2)!(3x)n(2n)!|
=|(3x)n+1(2n+2)!(2n)!(3x)n|
=|(3x)(2n+2)(2n+1)|
Now
L=limn|an+1an|
=limn|(3x)(2n+2)(2n+1)|
=|3x|limn|1n2(2+2n)(2+1n)|
L=0,
Here L=0<1 so series is absolutely convergent for all x.
Hence interval of convergence is (,).

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