Find the sum of the first 50 elements in for $x = 1.5$ $$\sum_n n x^n$$
Now, if I were to find an infinite convergant sum, I would know what to do, since $$\frac{1}{1-x} = \sum_nx^n$$ it's enough to differentiate this and perform some numerical manipulations.
However, as I am being asked to find the sum of a certain number of terms, I don't know how to proceed.
Hint: evaluate $$ x\partial_x\sum_n x^n $$ Also, as is well known, $$ \sum_{n = 0}^N x^n = \frac{1-x^{N+1}}{1-x} $$