Convergent series of nonnegative terms?

Question

If Σan is a convergent series of nonnegative terms, what can be said about Σ(n)a(n) and Σa(n) a(n+1)?

Answer 1

Let an = n^2. Σan converges Σ(n)(an) = Σ(1/n) doesn't.

In general you can apply the ratio test:

r = lim ((n+1)a(n+1))/((n)a(n) = a(n+1))/a(n)

if r < 1 the series converges, if r >1 it diverges, if r=1 it may or may not converge.


As for the second series for n large enough, an < 1 so (an)(an+1) < (an+1). By the comparison test, it will converge.

Answer 2

If n is finite it is also convergent.

Answer 3

They r homos? but i dont know who +1 is.