Determine the radius of convergence and the interval of convergence for each power series. sum_{n=1}^infty n!(2x-1)^n

sodni3

sodni3

Answered question

2020-11-30

Determine the radius of convergence and the interval of convergence for each power series.
n=1n!(2x1)n

Answer & Explanation

Clelioo

Clelioo

Skilled2020-12-01Added 88 answers

The given series is:
n=1n!(2x1)n
Applying Ratio test to the above series:
L=limn|an+1an|
=limn|(n+1)!(2x1)n+1n!(2x1)n|
=limn|(n+1)!(2x1)n(2x1)n!(2x1)n|
=limn|(n+1)(2x1)|
=|2x1|limn(n+1)
This means that L= provided x12
Hence, the given series only converges when x=12
Therefore, the radius of convergence = 0 and interval of convergence is x=12
Jeffrey Jordon

Jeffrey Jordon

Expert2021-12-27Added 2605 answers

Answer is given below (on video)

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