# Thread: Is the following series absolutely convergent or not

1. ## Is the following series absolutely convergent or not

Hi,

The series is : $\displaystyle \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{2n+1}$

The absolut series is : $\displaystyle \sum_{n=1}^{\infty} \frac{1}{2n+1}$

But I'm stuck at this very (easy ?) step : $\displaystyle s_{n} = \frac{1}{3} + \frac{1}{5} + \frac{1}{7} + ...$

But I can't determine, in my head, if it is convergent or not. I feel really dumb.

If I compare to the harmonic series, each term is lower.

The "quotient test" gives me : $\displaystyle \frac{2n+1}{2n+3} \textless 1$

What is the tiny thing I missed ?

The series is : $\displaystyle \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{2n+1}$
The absolut series is : $\displaystyle \sum_{n=1}^{\infty} \frac{1}{2n+1}$
Note that $\displaystyle \sum_{n=1}^{\infty} \frac{1}{2n+1}>\frac{1}{4}\sum_{n=1}^{\infty} \frac{1}{n}$