Find the values of x for which the given geometric series converges. Also, find the sum of the series (as a function of x) for those values of x. a) Find the values of x for which the given geometric series converges. b) Find the sum of the series sum_{n=0}^infty(-frac12)^n(x-5)^n

avissidep 2020-11-08 Answered
Find the values of x for which the given geometric series converges. Also, find the sum of the series (as a function of x) for those values of x.
a) Find the values of x for which the given geometric series converges.
b) Find the sum of the series
n=0(12)n(x5)n
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

Nathanael Webber
Answered 2020-11-09 Author has 117 answers

(a) Consider the given geometric series:
n=0(12)n(x5)n
Now, to find the value of x for which the series converges, apply ratio test by finding the following limit:
L=limn|an+1an| where an=(12)n(x5)n
=limn|(12)n+1(x5)n+1(12)n(x5)n|
=limn|(12)(x5)|
=|5x2|
Now, for the series to be convergent, the limit should be less than 1, that is:
L<1
|5x2|<1
1<5x2<1
2<5x<2
7<x<3
Now, check the convergence at 7 and 3:
At x=7
n=0(12)n(75)n=n=0(12)n2n=n=0(1)n
Which is divergent
At x=3
n=0(12)n(35)n=n=0(12)n(2)n=n=0(1)n
Which is divergent
Therefore, the required values of x for which the series converges is:
(b) Now, since, given series is a geometric series with:
a=1,r=(12)(x5)=(5x2)
Therefore, the sum of this infinite series will be given by:
S=a1r
=1(5x2)
=25x
Thus, required sum of series is:
S=25x Where 3

Not exactly what you’re looking for?
Ask My Question
Jeffrey Jordon
Answered 2021-12-16 Author has 2262 answers

Expert Community at Your Service

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