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!