Find the radius of convergence and interval of convergence of the series. sum_(n=1)^oo((-1)^n10^nx^n)/n^4

allhvasstH 2020-11-08 Answered
Find the radius of convergence and interval of convergence of the series.
n=1(1)n10nxnn4
You can still ask an expert for help

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question

Expert Answer

cyhuddwyr9
Answered 2020-11-09 Author has 90 answers

Here n=1(1)n10nxnn4=n=1an,where an=(1)n10nxnn4
Now |an+1an|=|10n+1xn+1n210nxn(n+1)2|
=|10x(nn+1)2|
This shows that
limn|an+1an|=|10x|limn(nn+1)2=10|x|
Therefore the given series converges if
10|x|<1|x|<110
and diverges if |x|>110. Therefore the radious of converges is R=110.When x=110 the the series beomes n=1(1)nn4 which converges by the Leibniz's test. Again when x=110 then the series becomes n=11n4 which converges by the p series test. Thereore the series converges converges in the interval |x|110

Not exactly what you’re looking for?
Ask My Question

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more