Unintuitive result for convergence of series

Question

Consider the series $\sum_1^{+\infty}\frac{1}{\sqrt{n}(n+1)}$. This infinite series converges and its sum is strictly smaller than 2.

Now consider the series $\sum_0^{+\infty}\frac{1}{2^n}$. It is a geometric series and is equal to 2.

Now I find this to be an incredibly unintuitive result. The terms of the second series decrease much more rapidly than those of the first series, and yet its sum is smaller. Is there a reason for this?

Answer 1

The initial terms in the series carry more weight than the later terms. The initial term ($n=0$) term in your first series is 1/2 and is 1 for your second series. This is why it is not surprising that the second sum is smaller.

Perhaps a more proper comparison is $\sum_{n=1}^\infty 1/(\sqrt{n}(n+1))$ and $\sum_{n=1}^\infty 2^{-n}$. The second sum converges to 1, which matches your intuition that it is smaller (now both sums have 1/2 as their $n=0$ term).

Answer 2

Please note that the first term of the first series is $\dfrac12$, whereas the first term of the second series is $1$. So, yes, the second series converges to a higher value, but it also starts from a higher value.