Calculating Convergent Series: Tips for $\sum_{n=1}^{\infty} n^2 w^n$

[cepheid]

Does anyone have tips on how to sum the following series?

$$\sum_{n=1}^{\infty} n^2 w^n$$

Region of convergence is for |w| < 1

 

[Galileo]

Try to exploit it's similarity with a geometric series.
Hint: Differentiate.

 

[xanthym]

Does anyone have tips on how to sum the following series?

$$\sum_{n=1}^{\infty} n^2 w^n$$

Region of convergence is for |w| < 1

$$(1) \ \ \ \ z \ = \ \sum_{n=0}^{\infty} w^{n} \ = \ (1 \ - \ w)^{-1} \ =$$

$$(2) \ \ \ \ \ \ \ \ \ \ = \ 1 \ + \ w \ + \ \sum_{n=2}^{\infty} w^{n}$$

$$3 \ \ \ \ \ \ \frac {dz} {dw} \ = \ \left ( 1 \ - \ w \right )^{-2} =$$

$$(4) \ \ \ \ \ \ \ \ \ \ \ \ \ \ = \ 1 \ \ + \ \ \sum_{n=2}^{\infty} n \cdot w^{n-1} \ = \ 1 \ \ + \ \ w^{-1} \cdot \sum_{n=2}^{\infty} n \cdot w^{n}$$

$$(5) \ \ \ \ \ \Longrightarrow \ \sum_{n=2}^{\infty} n \cdot w^{n} \ = \ w \cdot \left( \left(1 \ - \ w \right)^{-2} \ - \ 1 \ \right)$$

$$6 \ \ \ \ \ \ \frac {d^{2}z} {dw^{2}} \ = \ 2 \cdot \left ( 1 \ - \ w \right )^{-3} \ =$$

$$(7) \ \ \ \ \ \ \ \ \ \ = \ \sum_{n=2}^{\infty} n \cdot ( n \ - \ 1 ) \cdot w^{n-2} \ =$$

$$(8) \ \ \ \ \ \ \ \ \ \ = \ w^{-2} \cdot \sum_{n=2}^{\infty} n \cdot ( n \ - \ 1 ) \cdot w^{n} \ =$$

$$(9) \ \ \ \ \ \ \ \ \ \ \ \ \ \ = \ w^{-2} \cdot \left ( \sum_{n=2}^{\infty} n^{2} \cdot w^{n} \ - \ \sum_{n=2}^{\infty} n \cdot w^{n} \right ) \ =$$

$$(10) \ \ \ \ \ \ \ \ \ \ \ = \ w^{-2} \cdot \left ( \sum_{n=2}^{\infty} n^{2} \cdot w^{n} \ - \ w \cdot \left( \left(1 \ - \ w \right)^{-2} \ - \ 1 \ \right) \right ) \$$

$$(11) \ \ \ \ \color{red} \Longrightarrow \ \sum_{n=2}^{\infty} n^{2} \cdot w^{n} \ = \ 2 \cdot w^{2} \cdot \left ( 1 \ - \ w \right )^{-3} \ \ + \ \ w \cdot \left( \left(1 \ - \ w \right)^{-2} \ - \ 1 \ \right)$$

$$(12) \ \ \ \ \color{red} \Longrightarrow \ \sum_{n=1}^{\infty} n^{2} \cdot w^{n} \ \ = \ \ w \ \ + \ \ 2 \cdot w^{2} \cdot \left ( 1 \ - \ w \right )^{-3} \ \ + \ \ w \cdot \left( \left(1 \ - \ w \right)^{-2} \ - \ 1 \ \right)$$



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[spikemurphy]

Just Wanted To Know How To Make The Summation And Powers

 

[xanthym]

Just Wanted To Know How To Make The Summation And Powers

Try the following URL. Click on the actual formula or graphic for a pop-up window showing the "tex" code.
https://www.physicsforums.com/showthread.php?t=8997


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