Finding total elements in a series I have a confusion. 
How many terms in the following series are needed to make a sun greater than 5/2?
1/2 + 1/3 + 1/4 + 1/5 + 1/6 + .........

Is there any shortcut technique to find this out? 

Observe that, for $k\geq 1$,
\begin{align*}
\sum_{n=2}^{2^k}\frac{1}{n}&=\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}+\cdots+\frac{1}{2^k}\\
&>\frac{1}{2}+\frac{1}{4}+\frac{1}{4}+\frac{1}{8}+\frac{1}{8}+\frac{1}{8}+\frac{1}{8}+\cdots+\frac{1}{2^k} \\
&=\frac{1}{2}+\frac{1}{2}+\frac{1}{2}+\cdots+\frac{1}{2} \qquad (\textrm{k times}) \\
&=\frac{k}{2}.
\end{align*}
That is, 
$$ \sum_{n=2}^{2^k}\frac{1}{n}>\frac{k}{2} $$
Using this (crude) estimate, we need $2^5=32$ terms.
