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I am self teaching some basic analysis out of interest and I have a question on trying to prove a series diverges.

Question; Prove that (-1)^n (1+1/n) diverges.

My attempted approach was through contradiction:

Assume that (-1)^n (1+1/n) converges. Then for some ε>0, there exists an N such that for all n>N,

|(-1)^n (1+1/n) - L | < ε

Then I broke down into two cases; n is odd, or n is even

Suppose n is even, then we have | 1+1/n - L | < ε = 1 + 1/n < ε + L

Now suppose that n is odd, then we have | -1 - 1/n - L | < ε = -1 - 1/n < ε + L

Here is where I get stuck, I do not see where the contradiction is. Intuitively I know that the series jumps due to the (-1)^n and so alternately moves away from L, but I cannot show this.

If you guys could point me to where I am going wrong I would greatly appreciate it.

Thanks,

ZZD