How do you find the sum of the series 2/n from n=2 to 10?

Mar 19, 2017

Sum of the series is $5 \frac{1081}{1260}$

Explanation:

Such a series is called Harmonic series, which is just reciprocal of an arithmetic series.

Here the arithmetic series is $\frac{n}{2}$ and numbers are

$\left\{\frac{1}{2} , \frac{2}{2} , \frac{3}{2} , \frac{4}{2} , \ldots \ldots . , \frac{10}{2}\right\}$

and corresponding harmonic series (first $10$ terms) is

$\frac{2}{1} , \frac{2}{2} , \frac{2}{3} , \frac{2}{4} , \frac{2}{5} , \frac{2}{6} , \frac{2}{7} , \frac{2}{8} , \frac{2}{9} , \frac{2}{10}$

There is no formula for sum of first $n$ terms of a harmonic series and one has to do it manually or if desired in decimals, using a calculator or spreadsheet.

Sum of the series is $2 + 1 + \textcolor{red}{\frac{2}{3}} + \textcolor{b l u e}{\frac{1}{2}} + \textcolor{b r o w n}{\frac{2}{5}} + \textcolor{red}{\frac{1}{3}} + \frac{2}{7} + \textcolor{b l u e}{\frac{1}{4}} + \frac{2}{9} + \textcolor{b r o w n}{\frac{1}{5}}$

= $3 + \textcolor{red}{1} + \textcolor{b l u e}{\frac{3}{4}} + \textcolor{b r o w n}{\frac{3}{5}} + \frac{2}{7} + \frac{2}{9}$

= $4 + \frac{945 + 756 + 360 + 280}{1260} = 4 + \frac{2341}{1260} = 5 \frac{1081}{1260}$