Would the converse of this be true?

[Jamin2112]

Homework Statement

Theorem X. If the series ∑u_n is absolutely convergent, then each of the seires ∑a_n, ∑b_n is convergent, and ∑u_n = ∑a_n - ∑b_n. [...]

Homework Equations

Meh

The Attempt at a Solution

I was wondering if the converse is true: If I have a series ∑u_n, and if I can express it as the difference between two convergent sequences, then is ∑u_n convergent?

For example, one of the homework problems asks whether

1 - 1/3² + 1/5² - 1/7² + ...

is absolutely convergent.

Well, if I take the absolute value of those terms, I have

1 + 1/3² + 1/5² + 1/7² + ...

and I know ∑1/n² is convergent (from a previous section of the book), so of course any subsequence is convergent.

 

[micromass]

Yes, the converse is indeed true.
And you also solved your homework problem correctly!