Question

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

Answer 1

Sum of the series is `5 1081/1260`

Explanation:

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

Here the arithmetic series is `n/2` and numbers are

`{1/2,2/2,3/2,4/2,.......,10/2}`

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

`2/1,2/2,2/3,2/4,2/5,2/6,2/7,2/8,2/9,2/10`

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+color(red)(2/3)+color(blue)(1/2)+color(brown)(2/5)+color(red)(1/3)+2/7+color(blue)(1/4)+2/9+color(brown)(1/5)`

= `3+color(red)1+color(blue)(3/4)+color(brown)(3/5)+2/7+2/9`

= `4+(945+756+360+280)/1260=4+2341/1260=5 1081/1260`