Series a_n converges and a_n>=0. Does series e^{a_n}-1 converge as well?

I suppose sum {e^{a_n}} < e^{sum{a_n}} => the sequence of partial sums converges and the series should converge. But lim e^{a_n} /= 0!What is wrong ?

The same question about convergence for series n*a_{n^2}.

Thanks for any help=)