Does a Stated Condition Imply a Limit on s_k?

[utleysthrow]

Homework Statement

this is more of a question I had within a question... but here it is:

Suppose $$s_{k} = s_{2k-1} + s_{2k}$$

is true and I know for a fact that s_2k has no limit.

Would that imply that s_k has no limit as well? Or is that not enough?

Thanks in advance.

Homework Equations

The Attempt at a Solution

 

[Dick]

Not enough. Take s_k=(-1)^k.

 

[foxjwill]

Would that imply that s_k has no limit as well? Or is that not enough?

Actually, you've provided "too much"! See, the statement "If $$s_{2k}$$ has no limit, then neither does $$s_k$$" is the contrapositive of "If $$s_k$$ has a limit, then so does $$s_{2k}$$," the truth of which follows quite readily for all sequences from the definition of a sequence limit.

Not enough. Take s_k=(-1)^k.

But this doesn't satisfy the condition that $$s_{2k}$$ not converge.

 

[Dick]

But this doesn't satisfy the condition that $$s_{2k}$$ not converge.

Good point. You are right. If s_2k doesn't converge, s_k can't converge.