What is the convergence (absolute or conditional) of the alternate series: #f(x)=sum_{n=1} ^oo (-1)^n ((n)/(7n^2+3)) # ?

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Answer 1 · 2018-03-09T00:02:45

This series is conditionally convergent but not absolutely convergent.

Since this series is alternating, it will conditionally converge provided the individual terms tend to #0#, which they do:

#lim_(n->oo) (n/(7n^2+3)) = lim_(n->oo)1/(7n+3/n) = 0#

On the other hand, it is not absolutely convergent, since when #n >= 1# we have:

#7n^2+3 <= 10n^2#

So:

#n/(7n^2+3) >= n/(10n^2) = 1/10 1/n#

So:

#sum_(n=1)^oo abs((-1)^n (n/(7n^2+3))) >= 1/10 sum_(n=1)^oo 1/n = oo#