Proof of the nth term test

The nth term test basically says that if an infinite series SUM(a_n) converges, then lim a_n = 0.

The proof is almost immediate. Suppose our series converges to say S and let S_n denote the nth partial sum.

Then,

lim a_n = lim (S_n - S_(n-1)) = S-S = 0. That's it!

Note this is typically NOT how the nth term test is used. Usually in calculus books and in courses people use the contrapositive; that is,

IF lim a_n != 0 then the series SUM(a_n) diverges.