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This week in class, we covered one way to sum an arithmetico-geometric series. Now we're going to cover a different approach. Let $|r| < 1$, $$S = \sum_{k=0}^{\infty} r^k,$$and $$T = \sum_{k=0}^{\infty} k r^k.$$Our approach is to write $T$ as a geometric series in terms of $S$ and $r$. Give a closed form expression for $T$ in terms of $r$.

Aug 4, 2021

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Let

$f(x) = \sum_{k = 0}^\infty x^k = \frac{1}{1 - x}.$

Taking the derivative, we get

$f'(x) = \sum_{k = 1}^\infty kx^{k - 1} = \frac{x}{(1 - x)^2}.$

Then

$\sum_{k = 1}^\infty kx^k = \frac{x^2}{(1 - x)^2}.$

Therefore,

$\sum_{k = 1}^\infty kr^k = \frac{r^2}{(1 - r)^2}.$

Aug 4, 2021