Question

Does: sum (cos(npi)(n+ln(n)))/(2n+1) 1 to infinity converge, if so what tests need to be used?
Both the root test and the ratio test prove inconclusive.

Answer 1

alternates so maybe alternating series test, except it does not even look like the terms go to zero ....

Answer 2

doesn't converge i am pretty sure
it does alternate, but the terms don't go to zero

Answer 3

From the comparison test I have reduced it to being less than Cos(n*pi)/3 + (ln(n)Cos(n*pi))/3n
Maybe the alternating series test from there?

Answer 4

that stupid \(\cos(\pi n)\) is just a fancy way of saying \((-1)^n\)

Answer 5

Am I right in stating that |a_n| = (n+ln(n))/(2n+1) ?

Answer 6

yes, but that is not helpful
it diverges because the so called 'nth term' test does not hold
i.e. the limit of the terms is not zero

Answer 7

in fact the limit of the terms if \(\frac{1}{2}\) i think

Answer 8

Is that for a_n or |a_n|