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.
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.