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Let

S = 1 + r + r^{2} + r^{3} + \dots, | r | < 1

By considering
$\frac{d S}{d r}$
find a formula for
$1 + 2 r + 3 r^{2} + 4 r^{3} + \dots$
Hence, find the value of
$\sum_{k = 1}^{\infty} \frac{k}{2^{k - 1}}$

For (a), remember that

1 + r + r^{2} + \dots = \frac{1}{1 - r}

Again for (a), you can write this as

1 + r + r^{2} + \dots = (1 - r)^{- 1}

Differentiating the left side is easy; differentiating the right side needs the chain rule.

For (b), try writing out the first few terms of the sum and think about what value of $r$ in the formula you got from (a) would work this out for you.