true or false:

if the sum from n=1 to infinity of a(n) converges, and the sum from n=1 to infinity of b(n) diverges, then the series from n=1 to infinity of (a(n)*b(n)) diverges

i said this was true... is this correct?

I don't know.

What if
An = 1/x^5
and Bn = x^2
then An*Bn = 1/x^3

To determine whether the series from n=1 to infinity of (a(n)*b(n)) diverges when the sum from n=1 to infinity of a(n) converges and the sum from n=1 to infinity of b(n) diverges, we need to carefully analyze the situation.

Let's start by assuming that the sum from n=1 to infinity of a(n) converges to a finite value L, and the sum from n=1 to infinity of b(n) diverges (i.e., it goes to positive or negative infinity).

If possible, let's imagine there exists a finite value M such that the series from n=1 to infinity of (a(n)*b(n)) converges. By the definition of convergence, this would imply that the partial sums of the series (a(n)*b(n)) are bounded, and there exists N such that for all n > N:

| (a(1)*b(1)) + (a(2)*b(2)) + ... + (a(n)*b(n)) - M | < ε for some positive ε

Now, since the series from n=1 to infinity of b(n) diverges, we can find an index N1 such that for all n > N1, the partial sums of b(n) are greater than any given positive number, K:

b(N1+1) + b(N1+2) + ... + b(n) > K

Similarly, since the sum from n=1 to infinity of a(n) converges, we can find an index N2 such that for all n > N2:

| a(N2+1) + a(N2+2) + ... + a(n) - L | < ε/2 for some positive ε/2

Now, let's consider the partial sum of the series (a(n)*b(n)) for n > N = max(N1, N2). We can rewrite it as follows:

(a(1)*b(1)) + (a(2)*b(2)) + ... + (a(N-1)*b(N-1)) + (a(N)*b(N)) + (a(N+1)*b(N+1)) + ...

Since N is greater than both N1 and N2, we know that:

| (a(N2+1)*b(N2+1)) + (a(N2+2)*b(N2+2)) + ... + (a(N)*b(N)) | > K

Furthermore, for terms beyond N, we have:

| (a(N+1)*b(N+1)) + (a(N+2)*b(N+2)) + ... + (a(n)*b(n)) - M | < ε for some positive ε

Combining these inequalities, we have:

| (a(1)*b(1)) + (a(2)*b(2)) + ... + (a(N-1)*b(N-1)) + (a(N)*b(N)) + (a(N+1)*b(N+1)) + ... + (a(n)*b(n)) - M | > K - ε for some positive ε

Since ε can be chosen arbitrarily small, it follows that:

| (a(1)*b(1)) + (a(2)*b(2)) + ... + (a(n)*b(n)) - M | = ∞

This implies that the series from n=1 to infinity of (a(n)*b(n)) diverges.

Therefore, the statement is true: if the sum from n=1 to infinity of a(n) converges, and the sum from n=1 to infinity of b(n) diverges, then the series from n=1 to infinity of (a(n)*b(n)) diverges.