a) Find the values of x for which the given geometric series converges.

b) Find the sum of the series

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

$\sum _{n=0}^{\mathrm{\infty}}(-\frac{1}{2}{)}^{n}(x-5{)}^{n}$

a) Find the values of x for which the given geometric series converges.

b) Find the sum of the series

You can still ask an expert for help

Nathanael Webber

Answered 2020-11-09
Author has **117** answers

(a) Consider the given geometric series:

Now, to find the value of x for which the series converges, apply ratio test by finding the following limit:

Now, for the series to be convergent, the limit should be less than 1, that is:

Now, check the convergence at 7 and 3:

At x=7

Which is divergent

At x=3

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:

Therefore, the sum of this infinite series will be given by:

Thus, required sum of series is:

Jeffrey Jordon

Answered 2021-12-16
Author has **2262** answers

Answer is given below (on video)

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