Please help and provide a detailed solution conditionally co

Please help and provide a detailed solution.

conditionally, converges absolutely, or diverges.

Thanks in advance!

Determine whether the series converges

Solution

The series 1/(n+3) is divergent by the limit comparison test

The series 1/n^p is divergent whenever p<1

So 1/n=1/n^1/2 is divergent because 1/2<1

Now the limit comparison test implies that if you have two series a_n and bn

and lim a_n/b_n=1, then either both converge or both diverge

See http://en.wikipedia.org/wiki/Limit_comparison_test

Now lim (1/(n+3))/(1/n)=n/(n+3)=1, so 1/(n+3) diverges

However (-1)ⁿ/(n+3) converges by the alternating series test

The alternating series test says that if you have the series (-1)ⁿa_n

where a_n is a positive, monotone decreasing sequence , with limit 0, then the series converges

In our case 1/(n+3) is decreasing because 1/((n+1)+3)<1/n for any n

and the limit is 0.

So (-1)ⁿ/(n+3) converges but 1/(n+3) diverges. So it is conditionally convergent