TRUE OR FALSE

Question

1. if the lim as n->infinity of a(sub n)=0, then the sum from n=1 to infinity of a(sub n) converges

i said this was true because I know that if a (sub n) does NOT=0, it diverges

2. if the sum from n=1 to infinity of a(sub n) converges and a(sub n) does not =0, then the sun from n=1 to infinity of 1/(a(sub n)) diverges.
?????

Answer 1

1) is false. Counterexample: a_n = 1/n

Try to prove that sum from n=1 to infinity of 1/n is divergent.

"i said this was true because I know that if a (sub n) does NOT=0, it diverges"

Which is logically equivalent to:

Not divergent implies a_n ---> 0.

Of course, a convergent series must be such that a_n --->0. But the reverse is not true. So, the condition a_n ---> 0 is necessary but not sufficient for convergence.