calculus
posted by sarah on .
TRUE OR FALSE
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.
?????

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.