Would the converse of this be true?

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SUMMARY

The discussion centers on Theorem X, which states that if the series ∑un is absolutely convergent, then the series ∑an and ∑bn are also convergent, and ∑un equals ∑an minus ∑bn. The participant inquired whether the converse holds true: if a series ∑un can be expressed as the difference between two convergent sequences, does it imply that ∑un is convergent? The conclusion drawn is that the converse is indeed true, as demonstrated through the example of the series 1 - 1/32 + 1/52 - 1/72, which is shown to be absolutely convergent.

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Homework Statement



Theorem X. If the series ∑un is absolutely convergent, then each of the seires ∑an, ∑bn is convergent, and ∑un = ∑an - ∑bn. [...]

Homework Equations



Meh

The Attempt at a Solution



I was wondering if the converse is true: If I have a series ∑un, and if I can express it as the difference between two convergent sequences, then is ∑un 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.
 
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Yes, the converse is indeed true.
And you also solved your homework problem correctly!
 

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