Limit of a Sequence with (-1)n and Convergence Analysis

[Jamin2112]

Homework Statement

... I have (-1)ⁿ in a sequence and I'm trying to find the limit of that sequence?

Homework Equations

Obviously we have -1, 1, -1, 1, ...

The Attempt at a Solution

You see, I'm trying to consider the case x_n = n[1+(-1)ⁿ] + (1/n), to consider whether the sequence has just one accumulation point and is convergent.

 

[Tedjn]

The sequence does seem to have one accumulation point, but could it converge? Write out the first few terms.

 

[Jamin2112]

The sequence does seem to have one accumulation point, but could it converge? Write out the first few terms.

xn = {1, 9/4, 1/3, 65/8, 1/5, 125/12, 1/7, ...}

 

[HallsofIvy]

If a sequence is of the form $(-1)^n a_n$ where $a_n\ge 0$ for all n, then either the sequence does not converge or it converges to 0.

Proof: suppose it converged to a> 0. Let $\epsilon= a/2$. Then for any N> 0, there exist odd n> N so that $(-1)^na_n< 0$ from which $|a- (-1)^n a_n|> a$ and not less than $\epsilon= a/2$.

You do the case for a< 0.