Formula for the summation of the series $a+2a^2+3a^3+...$ upto nth term? I need a formula for the summation of the series $a+2a^2+3a^3+...$ upto nth term. Is there any way to simplify it?
You request $\sum_{j=1}^n j a^j$.  Then we have \begin{align*}
    \sum_{j=1}^n j a^j &amp;= a \sum_{j=1}^n j a^{j-1}  \\
        &amp;= a \sum_{j=1}^n \frac{\mathrm{d}}{\mathrm{d}a} a^j  \\
        &amp;= a \frac{\mathrm{d}}{\mathrm{d}a} \sum_{j=1}^n a^j  \\
        &amp;= a \frac{\mathrm{d}}{\mathrm{d}a} \frac{a^{n+1} - a}{a-1}  \\
        &amp;= a \frac{n a^{n+1} - n a^n - a^n + 1}{(a-1)^2}  \\
        &amp;= \frac{a^{n+1}(an - n -1) + a}{(a-1)^2}  \text{.}
\end{align*}
$\frac{(a\,*\,(1+na^{(1+n)}\,-\,(1+n)*a^n))}{(-1+a)^2}$

Thanks for the guide on how to use latex.