Sum of a series of the form n a^n

Asked Jun 14 '21 at 17:00

Active Jun 14 '21 at 17:24

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I was playing around with geometric series and I wondered if it was possible to find a formula for the sum of an altered geometric series (which actually stops being geometric) of the form:

$S_N = \sum_{n=0}^{N}na^n$

Under the condition $1>a>0$

Is there a way to compute this sum other than summing term by term? Can we find a formula that depends on $N$ and $a$ for this sum?

Thank you in advance!

edited Jun 14 '21 at 17:24

asked Jun 14 '21 at 17:00

João Miguel Ribeiro

Does this answer your question? Solve $\sum nx^n$ – Tortar Jun 14 '21 at 17:27
That other one is an infinite series, but this is a finite series. But the idea should be the same. – GEdgar Jun 14 '21 at 17:29
Hint: $$ na^n = a\frac{d}{{da}}a^n . $$ – Gary Jun 14 '21 at 17:35
There is a Wikipedia article To get a finite series you can subtract a short infinite sequence from a longer one. – Ross Millikan Jun 14 '21 at 17:36

Sum of a series of the form n a^n

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