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I don't understand why Rudin said The absolute convergence of (1) shows that the computation is correct here, I think I can do this computation without absolute convergence?

How to Multiply Two Infinite Series Correctly?

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    rearranging the summands (first equal sign). – peek-a-boo Jun 18 '23 at 04:31
  • @peek-a-boo Sorry, I still not quite understand why I need that property? –  Jun 18 '23 at 04:38
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    Look up the Riemann rearrangement theorem. This is all standard stuff covered in analysis texts, and since you mention Rudin, look at PMA, particularly theorems 3.50, 3.54 (in fact, just read everything from the section on absolute convergence onwards). – peek-a-boo Jun 18 '23 at 04:41
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    In fact, if you think this is so easy to do without absolute convergence, try to prove it. If you manage to give a “proof”, then go read what I mentioned above, and then find the mistake in your “proof”. This will be a very useful exercise in gaining familiarity with limits, series and definitions. – peek-a-boo Jun 18 '23 at 04:44
  • I like this video @TerryYoung. There is a brief discussion of what you are asking (among many other things): https://www.youtube.com/watch?v=JOg9kQp7pig&list=PLw3pvR_YJeRcMaubDZvkjayqDJT4Tx47A&index=4 – Andrew Jun 18 '23 at 04:51

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