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Converge or Diverge?

1.

A(n+1) = An + 1/(5^n) (recursive sequence)

That should read a sub n plus one equals a sub n plus one over 5 to the n

I believe it converges, but would like to be sure.

2. sigma from 1 to infinity of:

(1+5/n)^n

That should read the summation from n equals 1 to infinity of one plus 5 over n quantity to the n-th power. I believe this diverges, but would like to be sure.

Thank you in advance!