The sum to infinity of the following series 2 | vedclass

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(C) The given series is $S = 2 + \frac{1}{2} + \frac{1}{3} + \frac{1}{2^2} + \frac{1}{3^2} + \frac{1}{2^3} + \frac{1}{3^3} + \dots \infty$.We can rewr

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$7/2$

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(C) The given series is $S = 2 + \frac{1}{2} + \frac{1}{3} + \frac{1}{2^2} + \frac{1}{3^2} + \frac{1}{2^3} + \frac{1}{3^3} + \dots \infty$.We can rewrite this series by grouping the terms with powers of $1/2$ and $1/3$ separately:$S = (1 + \frac{1}{2} + \frac{1}{2^2} + \frac{1}{2^3} + \dots \infty) + (1 + \frac{1}{3} + \frac{1}{3^2} + \frac{1}{3^3} + \dots \infty)$.Note that the original series had a $2$ at the start,which is $1 + 1$. We distribute these two $1$s into the two geometric series.Both series are infinite geometric series with the sum formula $S = \frac{a}{1-r}$.For the first series: $a = 1, r = 1/2$. Sum $= \frac{1}{1 - 1/2} = \frac{1}{1/2} = 2$.For the second series: $a = 1, r = 1/3$. Sum $= \frac{1}{1 - 1/3} = \frac{1}{2/3} = 3/2$.Total sum $= 2 + 3/2 = 7/2$.

The sum to infinity of the following series $2 + \frac{1}{2} + \frac{1}{3} + \frac{1}{2^2} + \frac{1}{3^2} + \frac{1}{2^3} + \frac{1}{3^3} + \dots$ will be

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