Question

If a_n >0 and b_n >0 and series ∑ sqrt( (a_n)^2 +(b_n)^2 ) converges, then ∑a_n and ∑b_n both converge.

True or false? If true, why? If false, give a counterexample.

Answer 1

The harmonic series is often useful in cases like this.

1 + 1/2 + 1/3 + 1/4 + ... diverges.

1 + 1/4 + 1/9 + 1/16 + ... converges.

I think you can see from this how to make it clear that the assertion is false.