# What is the sum 1+1/3+1/5+1/7+1/9+1/11+... ?

Nov 27, 2017

Like the harmonic series, this sum does not converge.

#### Explanation:

This sum does not converge.

$1 + \textcolor{red}{\frac{1}{3}} + \textcolor{\mathmr{and} a n \ge}{\frac{1}{5} + \frac{1}{7}} + \textcolor{g r e e n}{\frac{1}{9} + \frac{1}{11} + \frac{1}{13} + \frac{1}{15}} + \textcolor{p u r p \le}{\frac{1}{17} + \frac{1}{19} + \frac{1}{21} + \frac{1}{23} + \frac{1}{25} + \frac{1}{27} + \frac{1}{29} + \frac{1}{31}} + \textcolor{b l u e}{\frac{1}{33} +} \ldots$

$> 1 + \textcolor{red}{\frac{1}{4}} + \textcolor{\mathmr{and} a n \ge}{\frac{1}{8} + \frac{1}{8}} + \textcolor{g r e e n}{\frac{1}{16} + \frac{1}{16} + \frac{1}{16} + \frac{1}{16}} + \textcolor{p u r p \le}{\frac{1}{32} + \frac{1}{32} + \frac{1}{32} + \frac{1}{32} + \frac{1}{32} + \frac{1}{32} + \frac{1}{32} + \frac{1}{32}} + \textcolor{b l u e}{\frac{1}{64} +} \ldots$

$= 1 + \textcolor{red}{\frac{1}{4}} + \textcolor{\mathmr{and} a n \ge}{\frac{1}{4}} + \textcolor{g r e e n}{\frac{1}{4}} + \textcolor{p u r p \le}{\frac{1}{4}} + \ldots$

does not converge