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(x−5)n

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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(x−5)n

Nathanael Webber · Accepted Answer

(a) Consider the given geometric series: ∑n=0∞(−12)n(x−5)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(x−5)n =limn→∞|(−12)n+1(x−5)n+1(−12)n(x−5)n| =limn→∞|(−12)(x−5)| =|5−x2| Now, for the series to be convergent, the limit should be less than 1, that is: L&amp;lt;1 |5−x2|&amp;lt;1 −1&amp;lt;5−x2&amp;lt;1 −2&amp;lt;5−x&amp;lt;2 −7&amp;lt;−x&amp;lt;−3 Now, check the convergence at 7 and 3: At x=7 ∑n=0∞(−12)n(7−5)n=∑n=0∞(−12)n2n=∑n=0∞(−1)n→∞ Which is divergent At x=3 ∑n=0∞(−12)n(3−5)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)(x−5)=(5−x2) Therefore, the sum of this infinite series will be given by: S=a1−r =1(5−x2) =25−x Thus, required sum of series is: S=25−x Where 3

Jeffrey Jordon · Accepted Answer

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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

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