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, $$ \sum_{n = 0}^N x^n = \frac{1-x^{N+1}}{1-x} $$

  • $\begingroup$ The second part of your answer actually solved my problem, thank you! What does this symbol $\partial_x$ mean in the first part? $\endgroup$ – Aemilius Nov 14 '18 at 11:07
  • $\begingroup$ @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 $$\sum_{n=0}^{50} \mathbf{n }x^n$$ $\endgroup$ – denklo Nov 14 '18 at 11:08
  • $\begingroup$ @Aemilius or is this a typo? $\endgroup$ – denklo Nov 14 '18 at 11:10
  • $\begingroup$ How did you come up with the idea to use the sum containing the partial derivative? Intuition? $\endgroup$ – Aemilius Nov 14 '18 at 11:10
  • $\begingroup$ Yes, but I can differentiate what you have written and obtain my result, or at least I guess so. $\endgroup$ – Aemilius Nov 14 '18 at 11:11

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