coverges find limit?

Question

Coverges or diverges? If

(n with a is subscript)
an= 1+(-1)^n

Answer 1

If n is odd, then (-1)^n = -1 and if n is even, then (-1)^n = 1 .Therefore, for odd n a_n = 0 and, for even n, a_n = 2. It follows the terms of (a_n) are 0, 2, 0, 2, 0, 2..... and the sequence diverges. We can observe this sequence has a subsequence that converges to 0 and another subsequence that converges to 2.

Answer 2

a1 = 0
a2 = 2
a3 = 0
a4 = 2
etc.

Convergence requires that for a high enough n, everything past (an) will be as close as you want to the limit. Clearly this isn't the case.