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

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Answer 1 · 2017-04-17T17:44:09

By Divergence Test.

Please see the details below.

Let us consider:

$lim_(n to infty)n/(2n+1)$

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

$=lim_(n to infty)1/(2+1/n)=1/(2+0)=1/2$ ,

which means that

$lim_(n to infty)((-1)^(n-1)n)/(2n+1)$ does not exist. ( $ne0$ )

By Divergence Test, the given series diverges.

I hope that this was clear.

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