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Why is $$\sum_{k=0}^\infty \frac{k}{(1+r)^k} > 1/r$$ where $r$ is interest rate (small number)

Can someone give me a hint?

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The Geometric series formula, for $-1<q<1$: $$ \sum_{k=0}^{\infty}q^k=\frac{1}{1-q} $$ Derive by q $$ \sum_{k=0}^{\infty}kq^{k-1}=\frac{1}{(1-q)^2} $$ Multiply by q $$ \sum_{k=0}^{\infty}kq^{k}=\frac{q}{(1-q)^2} $$ Substitute $q=\frac{1}{1+r}$ $$ \sum_{k=0}^{\infty}\frac{k}{(1+r)^{k}}=\frac{1+r}{r^2}=\frac{1}{r^2}+\frac{1}{r} >\frac{1}{r}$$

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If $|x|<1$, then $$ \sum_{k=0}^{\infty}kx^k=\frac{x}{(1-x)^2} $$

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