Sunday, November 15, 2015

Convergence of the series sumln(1+frac(1)nn+1)

I want to show that the series whose nth term is
an=ln(1+(1)nn+1) is convergent. I wanted to use the limit comparison test to compare it to the p series but an is not positive. I thought of writing the power series representation of an using the power series representation of ln(1+x) with x=bn=(1)nn+1 we find that
an=bn12b2n+13b3n14b4n+
Now the seris bn is convergent by the alternating series test and the other terms are all terms of absolutely convergent series but it is an infinte sum, can I say so ? I mean is the infinite sum of convergent series a convergent series ? Is this correct and is there any other way to do it ?

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