Thread: Adv. Calc Infinite Series of Real Numbers

Prove that if ∑ (k=1 to ∞) a_k converges, then its partial sums s_n are bounded.

Show that the converse in not true or that the series may have bounded partial sums and still diverge.

Thanks.

Originally Posted by taypez

Prove that if ∑ (k=1 to ∞) a_k converges, then its partial sums s_n are bounded.

Show that the converse in not true or that the series may have bounded partial sums and still diverge.

Thanks.

That is easy. What does it mean convergence of a series? It means the sequence of partial sums converges. But convergent sequences are bounded. Finished.

So can I say in proof:

Let the sum (k=1 to infinity) a_k be convergent. Then the sequence of partial sums converge by definition. Then for every epsilon >0 there is an N in N s.t. n>=N implies that abs (s_n -s) <epsilon. So therefore the partial sums are bounded.

Sorry, this is the part I'm not good at.