Can two different infinite series converge to the same limit?

[raphile]

Hi,

The title of the thread doesn't adequately describe the question I want to ask, so here it is:

Suppose we have two infinite series, \sum_{n=1}^{\infty}a_n and \sum_{n=1}^{\infty}b_n, both of which are convergent. Also suppose a_n \leq b_n for all n, and a_n < b_n for at least one n. Is it possible that they both converge to the same limit? Or can we say that \sum_{n=1}^{\infty}a_n is strictly smaller than \sum_{n=1}^{\infty}b_n?

If the answer to the above question is that they can have the same limit, then what if a_n<b_n for infinitely many n?

Thanks!

 

[jgens]

Suppose a_i < b_i and that a_n \leq b_n for all n \neq i. Then if \sum_{n \in \mathbb{N}}a_n and \sum_{n \in \mathbb{N}}b_n converge, it is easy to show the following:
\sum_{n \leq i}a_n < \sum_{n \leq i}b_n
\sum_{n > i}a_n \leq \sum_{n > i}b_n
Adding these together gives
\sum_{n \in \mathbb{N}}a_n = \sum_{n \leq i}a_n + \sum_{n > i}a_n < \sum_{n \leq i}b_n + \sum_{n > i}b_n = \sum_{n \in \mathbb{N}}b_n
This should answer your question.

 

[mathman]

There is no way they can have the same limit. If you compare finite sums including at least one n where a_n < b_n, then the a partial sum will be smaller and there is no way it can make up the difference without a term with a > b.

 

[raphile]

Ok, that's great, thanks!