Occasionally when teaching the sequences and series material in second-semester calculus I’ve included the following question as a bonus:

**Question**: Suppose is absolutely convergent. Does that imply anything about the convergence of ?

The answer is that converges. I’m going to give two proofs. The first is the more straightforward approach, but it is somewhat longer. The second is clever and shorter.

**First method**: If is absolutely convergent, then, by the divergence test, . Thus there exists some such that if then . This means that, for , . By the direct comparison test, then, converges.

**Second method**: As we argued above, . Thus . By the limit comparison test, then, converges.

(The first method is David Mitra’s answer and the second method is my answer to this Math.SE question.)

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I admire your work, you pick very nice problems! I just want to ask something. How are you writing formulas? Please don’t tell me that you generate them as pics and insert in the code.. I think there is WP plugin that works with MathJax so that you write code in latex and it is rendered nicely.

I use the LaTeX environment in WordPress.

I am sorry then. You can delete my comments if you wish.

No need to apologize. And thanks for your interest in my work!

How about this (assuming the terms of the series positive so I can avoid absolute value signs): For all , . Thus where . Thus the sequence of partial sums of is increasing and bounded above by so the series converges.

Nice, Tom!