calculus

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.
?????

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  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.

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