Question

Write out the series given by:

Answer 1

When $n=1$, we have

$\dfrac n{2n+1}=\dfrac1{2\cdot1+1}=\dfrac13$

When $n=2$, we have

$\dfrac n{2n+1}=\dfrac2{2\cdot2+1}=\dfrac25$

and so on. The series is then

$\displaystyle\sum_{n=1}^6\frac n{2n+1}=\frac13+\frac25+\frac37+\frac49+\frac5{11}+\frac6{13}$