# Find the limit to the following

Please if any one can find the following limit:

Let $b$ be a real number, $b>1$. Compute $$\lim_{n\to\infty}n\cdot\biggl(\frac1{b^n}+\frac1{b^{2n}}+\frac1{b^{3n}} +\dots+\frac1{b^{(n-1)n}}+\frac1{b^{n^2}}\biggr).$$

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Let's squeeze! $$0 \leq n\left(\frac{1}{b^n} + \dots + \frac{1}{b^{n^2}}\right) \leq n\times \frac{n}{b^n} \xrightarrow[n\to\infty]{} 0$$
@Ju'x: shouldn't there be only $n^2$ on the right hand side of the inequality (instead of $n^3$)? There are only $n$ terms between the parentheses. – Matt L. Apr 12 '13 at 8:49