Calculating $\sum_{k=1}^\infty \frac{k^2}{2^k}=\frac{1}{2}+\frac{4}{4}+\frac{9}{8}+\frac{16}{16}+\frac{25}{32}+\cdots+\frac{k^2}{2^k}+\cdots$

Question

I want to know the value of $$\sum_{k=1}^\infty \frac{k^2}{2^k}=\frac{1}{2}+\frac{4}{4}+\frac{9}{8}+\frac{16}{16}+\frac{25}{32}+\cdots+\frac{k^2}{2^k}+\cdots$$ I added up to $k=50$ and got the value $5.999999999997597$, so it seems that it converges to $6.$ But, I don't know how to get the exact value. Is there any other simple method to calculate it?

Answer 1

If we start with the power series $$ \sum_{k=0}^{\infty}x^k=\frac{1}{1-x} $$ (valid for $|x|<1$) and differentiate then multiply by $x$, we get $$ \sum_{k=1}^{\infty}kx^k=\frac{x}{(1-x)^2}$$ If we once again differentiate then multiply by $x$, the result is $$ \sum_{k=1}^{\infty}k^2x^k=\frac{x(x+1)}{(1-x)^3}$$ and setting $x=\frac{1}{2}$ shows that $$ \sum_{k=1}^{\infty}k^22^{-k}=\frac{\frac{3}{4}}{\frac{1}{8}}=6 $$ as you guessed.

Answer 2

Start with the geometric series $\frac{1}{1-x} = \sum_i x^i$. Differentiate it once to get $\frac{d}{dx} \left[ \frac{1}{1-x} \right] = \frac{d}{dx} \sum_i x^i = \sum_i i x^{i-1}$. Differentiate again to get $\frac{d^2}{dx^2} \left[ \frac{1}{1-x} \right] = \frac{d^2}{dx^2} \sum_i x^i = \sum_i i (i-1) x^{i-2}$.

Now, plug in $x= \frac{1}{2}$ and adjust indices in teh summations.

Answer 3

Alternatively, note that if $T = \sum_{k \geq 0}k^2 2^{-k}$ then $$T = 2T-T = \sum_{k\geq 0}\frac{2k+1}{2^k} = 2\sum_{k \geq 0}\frac{k}{2^k} + \sum_{k \geq 0}2^{-k} $$

But we can let $S= \sum_{k \geq 0}k2^{-k}$ and note that $$S = 2S-S = \sum_{k\geq 0}2^{-k} = 2.$$

Hence $T = 2\cdot 2 + 2 = 6.$