How do you simplify (1/(1+i)) + (1/(2+i)) + (1/(3+i)) ?

Mar 19, 2016

$\frac{1}{1 + i} + \frac{1}{2 + i} + \frac{1}{3 + i} = \frac{6}{5} - \frac{4}{5} i$

Explanation:

First, note that for any $n$, we have

$\frac{1}{n + i} = \frac{n - i}{\left(n + i\right) \left(n - i\right)} = \frac{n - i}{{n}^{2} + 1}$

If we substitute in $n = 1 , 2 , 3$ we get

$\frac{1}{1 + i} + \frac{1}{2 + i} + \frac{1}{3 + i} = \frac{1 - i}{2} + \frac{2 - i}{5} + \frac{3 - i}{10}$

$= \frac{5 \left(1 - i\right)}{10} + \frac{2 \left(2 - i\right)}{10} + \frac{3 - i}{10}$

$= \frac{5 - 5 i + 4 - 2 i + 3 - i}{10}$

$= \frac{12 - 8 i}{10}$

$= \frac{12}{10} - \frac{8}{10} i$

$= \frac{6}{5} - \frac{4}{5} i$