Write out the first few terms of each series to show how the series start. Then find out the sum of the series. ∑n=0∞(52n+13n)

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Write out the first few terms of each series to show how the series start. Then find out the sum of the series.  ∑n=0∞(52n+13n)

Macsen Nixon · Accepted Answer

Consider the given series as I=∑n=0∞(52n+13n) To find the few terms ,substitute the value of n from 0 to infinity in given series i.e. I=(520+130)+(521+131)+(522+132)+... =(5+1)+(15+26)+(45+436)+... =6+176+4936+... Since the given series is geometric series. So, use formula of geometric sum of infinite series Sn=a(1−rn)1−r where a is the first term of series and r is geometric ratio between every two terms. As, the given series can be written as I=∑n=0∞(52n)+∑n=0∞(13n) If the series ∑an=∑n=0∞(52n)and∑bn=∑n=0∞13n are convergent series then it can be written in the form of ∑an+bn=∑an+∑bn Check the convergence of the series by ratio test limn→∞an+1an=limn→∞52n+152n =limn→∞2n2n+1 =limn→∞2n2n(2) =limn→∞an+1an=12&amp;lt;1 As, the limit for above series is less than 1 . Therefore, the above series is convergent. Now, check for another series =limn→∞bn+1bn=13n+113n =limn→∞3n3n(3) =limn→∞bn+1bn=13&amp;lt;1 It implies that the limit of above series is less than 1 and hence is convergent. Therefore, the given series is I=5∑n=0∞(12n)+∑n=0∞(13n) By using formula of sum of infinite series 5(2)+32 232 Hence,the sum of given infinite series is 232

Jeffrey Jordon · Accepted Answer

Answer is given below (on video)

Write out the first few terms of each series to show how the series start. Then find out the sum of the series. sum_{n=0}^infty(frac{5}{2^n}+frac{1}{3^n})

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