Show that the series converges. What is the value of the series? ∑n=2∞(−53)n(25)n+1

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Show that the series converges.  What is the value of the series?  ∑n=2∞(−53)n(25)n+1

ensojadasH · Accepted Answer

We have to Show that the series converges. What is the value of the series? Series is given below: ⇒∑n=2∞(−53)n(25)n+1 We will use geometric series test . Geometriic series test says if series of the form arn−1 and |r|&amp;lt;1 then we can say that series is converges and sum of series is given by S. Work is shown below: ⇒∑n=1∞arn−1 |r|&amp;lt;1 ⇒S∞=a1−r Now compare the given series to geometric series and find value of r . If r&amp;lt;1 then series converges. In this case r=23&amp;lt;1 so series converge. After that find value of a (starting term of series). With the help of a, r and sum of geometric series formula we will find sum. Work is shown below: ⇒∑n=2∞(−53)n(25)n+1 ⇒∑n=2∞(−53)n(25)n+1 ⇒∑n=2∞25(−1)n(23)n ⇒|r|=|−23| |r|=23&amp;lt;1 Previous step continue ⇒n=2∞25(−1)n(23)n ⇒845−16135+32405... ⇒a=845 ⇒r=−23 ⇒S∞=8451+23 =875 Answer ⇒S∞=875

Jeffrey Jordon · Accepted Answer

Answer is given below (on video)

Show that the series converges. What is the value of the series? sum_{n=2}^infty(-frac{5}{3})^n(frac{2}{5})^{n+1}

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Answer & Explanation

New Questions in Calculus 2