# Sum of the first 50 elements of the sum $\sum n a^n$

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.

## 1 Answer

Hint: evaluate $$x\partial_x\sum_n x^n$$ Also, as is well known, $$\sum_{n = 0}^N x^n = \frac{1-x^{N+1}}{1-x}$$

• The second part of your answer actually solved my problem, thank you! What does this symbol $\partial_x$ mean in the first part? – Aemilius Nov 14 '18 at 11:07
• @Aemilius, its the derivative wrt. x. Are you sure that you only need the second part? In your question the sum to evaluate is not the geometirc series, but $$\sum_{n=0}^{50} \mathbf{n }x^n$$ – denklo Nov 14 '18 at 11:08
• @Aemilius or is this a typo? – denklo Nov 14 '18 at 11:10
• How did you come up with the idea to use the sum containing the partial derivative? Intuition? – Aemilius Nov 14 '18 at 11:10
• Yes, but I can differentiate what you have written and obtain my result, or at least I guess so. – Aemilius Nov 14 '18 at 11:11