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I have the series $a_n(x) = x^n \dot{} a (b^n - c^n)$ where a b and c are constants. I want to find the sum of $$\sum_{n=1}^{\infty} a_n(x)$$ for any value of x (less than one of course). I learned in algebra 2 how to find the infinite sum for just $x^n$ for $x<1$, but I have yet to learn to solve more complex infinite sums like this. How would I do this?

By the way I know there's some easy ways to get a close approximate decimal answer, but I'm looking for a way that finds the exact answer, like how $\zeta (2) = \frac{\pi^2}{6} $ exactly.

Ryan
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  • You'll need conditions on $b$ and $c$. Then $a\cdot x^n(b^n-c^n)=a\cdot[(bx)^n-(cx)^n]$. Now you can "distribute" the sum and sum the series separately. – Clayton May 17 '17 at 21:00

1 Answers1

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For $x $ such that

$|x|<\frac {1}{\max (|b|,|c|)} $ the sum of geometric series gives

$$ax\left(\frac {b}{1-xb}-\frac {c}{1-xc}\right) $$

Davide Giraudo
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hamam_Abdallah
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