Determine the radius and interval of convergence of the following power series. sum_{k=0}^infty(2x)^k

Dottie Parra

Dottie Parra

Answered question

2021-02-05

Determine the radius and interval of convergence of the following power series.
k=0(2x)k

Answer & Explanation

Leonard Stokes

Leonard Stokes

Skilled2021-02-06Added 98 answers

Given power series is,
k=0(2x)k
To find :radius of convergence& interval of convergence.
We use root test here to find radius and interval of convergence.
The root test is given as,
If L=limn|an|n=limn|an|1n
if L<1, then the series converges absolutely.
if L>1, then the series diverges.
if L=1, then the series is either convergent or divergent.
Here ak=(2x)k
we apply root test here for ak
Let L=limk(ak)k=limk(ak)1k
By putting value of ak we get,
L=limk(ak)1k=limk((2x)k)1k
=limk(2x)
=2x
The series converges if |L|<1
|2x|<1
2|x|<1
|x|<12

If |xa| then R is radius of convergence and the interval of convergence is (aR,a+R)

Here |x|<12 therefore a=0
Therefore radius of convergence R=12
and the interval of convergence is (012,0+12)=(12,12)
Therefore radius of convergence =12
and the interval of convergence is (12,12)

Jeffrey Jordon

Jeffrey Jordon

Expert2021-12-27Added 2605 answers

Answer is given below (on video)

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