convergent or divergent series?

Question

Let of ∑an when n goes 1 to infinity and ∑bn when n goes from 1 to infinity be convergent series with positive terms. Which of these series MUST converge? ∑(an + bn)? ∑(anbn)? ∑(an / bn)? or all of them?

Answer 1

Given:
∑a(n) converges
∑b(n) converges

Claim: ∑(a(n) + b(n)) converges.
Proof: Since ∑a(n) converges, let's assign a value to it; A.
Since ∑b(n) converges, let's assign a value to it; B.

A + B =
∑a(n) + ∑b(n) =
[ (a1 + a2 + ... ) + (b1 + b2 + .... ) ] =
[ (a1 + b1) + (a2 + b2) + (a3 + b3) + ... ] =
∑[ (a(n) + b(n) ], which demonstrates that this converges.

I'm not immediately certain of ∑[ (a(n) b(n) ].

But ∑ [ a(n)/b(n) ], this does not necessarily converge.

Let a(n) = 1/n^3 and b(n) = 1/n^2. It is clear that ∑a(n) and ∑b(n) converge, because they are both p-series. However,

a(n)/b(n) = [ (1/n^3) / (1/n^2) ] = [ n^2 / n^3 ] = (1/n), and
∑ (1/n) is the harmonic series, which diverges.

Answer 2

∑(an + bn) must converge.

∑anbn, if ∑an^2 and ∑bn^2 each converge; but that is not stated.

Answer 3

an*bn if both converge lim an = lim bn = 0 so an and bn <1
so an *bn