How do you show that the series 1/3-2/5+3/7-4/9+...+((-1)^(n-1)n)/(2n+1)+... diverges?

Apr 17, 2017

By Divergence Test.

Explanation:

Let us consider:

${\lim}_{n \to \infty} \frac{n}{2 n + 1}$

By dividing the numerator and the denominator by $n$,

$= {\lim}_{n \to \infty} \frac{1}{2 + \frac{1}{n}} = \frac{1}{2 + 0} = \frac{1}{2}$,

which means that

${\lim}_{n \to \infty} \frac{{\left(- 1\right)}^{n - 1} n}{2 n + 1}$ does not exist. ($\ne 0$)

By Divergence Test, the given series diverges.

I hope that this was clear.