Question

The sequence (-1)^(n)*e^(1/n) is divergent or convergent?

Answer 1

Do the terms approach zero (0)?

Answer 2

Well, (1/n) approaches zero when n tends to infinite.

Answer 3

The term (-1)^n makes it just float sign every n, so the interesting factor is e^1/n.
Remember how both 1/n and e^something behaves. If n diverges, 1/n tends to zero. Now, if "something" approaches zero, how does e^"something" behaves?

Answer 4

Well it behaves like e^0 which is 1 I think.

Answer 5

True. Now what does the "jumping sign" term do to that 1?

Answer 6

No one should care about the oscillating sign. There terms do not disappear (tend toward zero). They approach 1 (not zero). It will not ever converge without zero.

Answer 7

Changes it to -1 if n is odd or 1 is n is even.

Answer 8

Then it doesn't neither converge nor diverge.

Answer 9

Well thanks for the help, Paounn.