I need help with these. Decide whether or not each of the following statements is true all of the time. If it's true, write an explanation or a proof. If it's false, give a counterexample. Assume (a sub n) and (b sub n) > 0 for all n.
If summation 1 to infinity of (a sub n) = s, then summation 1 to infinity of (1/ a sub n) = 1/s.
If summation (a sub n) is a convergent series, then so is summation ((a sub n)^2).
If summation (a sub n) and summation (b sub n) are both divergent series where (a sub n) doesn't equal (b sub n), then so is the summation (a sub n - b sub n).
If summation [(a sub n)/(b sub n)] is a divergent series, then summation (b sub n) must be a divergent series.
If summation 1 to infinity of (a sub n) = A and summation 1 to infinity of (b sub n) = B, then summation 1 to infinity of [(a sub n)*(b sub n)] = AB.
I thought the first one was false and I gave the example (a sub n) = square root of n.
I guessed that the last one is false because you'd be multiplying the numbers of the series with summation [(a sub n)*(b sub n)] and with AB, you be multiplying the overall sum. I don't know if I'm right though.
Anyways, I did 15 of them and I'm now stuck on these 5. Thanks a lot guys and gals!