Intuition behind the rationality of some infinite sums of powers over powers

Question

Recently I learned that some sums of the form

$$\sum_{n=0}^\infty \frac {n^a} {b^n}$$

have rational or indeed integer values. For example,

$$\sum_{n=0}^\infty \frac {n^3} {2^n} = 26$$

and

$$\sum_{n=0}^\infty \frac {n^4} {3^n} = 15$$

and

$$\sum_{n=0}^\infty \frac {n^2} {3^n} = {\frac 3 2} $$

The blog post I link to above justifies some general results, but it feels to me as though for some of the smaller-number cases there might be 'nice' simple demonstrations, perhaps appealing to some geometrical intuition. Are there any such?

_{not sure about tags, please refine if you know better!}

Answer 1

There's a nice, simple demonstration for $\sum_1^{\infty}{n\over a^n}$ for $|a|>1$: $$ {1\over a}+{2\over a^2}+{3\over a^3}+{4\over a^4}+\dotsb=({1\over a}+{1\over a^2}+{1\over a^3}+{1\over a^4}+\dotsb)+({1\over a^2}+{1\over a^3}+{1\over a^4}+\dotsb)+({1\over a^3}+{1\over a^4}+\dotsb)$$

$$+\dotsb={1\over a-1}+{1\over(a-1)a}+{1\over(a-1)a^2}+\dotsb={a\over(a-1)^2} $$

All it takes is lots of use of the formula for the sum of a geometrix series.