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 November 6th, 2019, 10:19 AM #1 Newbie   Joined: Jan 2014 Posts: 20 Thanks: 0 absolutely convergent series Hello. I want to show that, for an absolutely convergent series $\displaystyle \sum_{n=1}^{\infty}a_n$, we have $\displaystyle \left|\sum_{n=1}^{\infty}a_n\right|\leq\sum_{n=1}^ {\infty}|a_n|$. Let $\displaystyle M$ be an positive integer. I begin with the triangle inequality $\displaystyle \left|\sum_{n=1}^{M}a_n\right|\leq\sum_{n=1}^{M}|a _n|$ and taking limit of both sides as $\displaystyle M\to\infty$. How to show that $\displaystyle \lim_{M\to\infty}\left|\sum_{n=1}^{M}a_n\right| = \left|\sum_{n=1}^{\infty}a_n\right|$? Thank you. Last edited by skipjack; November 6th, 2019 at 04:32 PM. November 6th, 2019, 10:49 AM #2 Senior Member   Joined: Dec 2015 From: Earth Posts: 823 Thanks: 113 Math Focus: Elementary Math $\displaystyle |a_n - L| < \epsilon \rightarrow 0$. $\displaystyle |x+y|\leq |x|+y \leq |x|+|y|$. For n-variables : $\displaystyle |x_1 + ... +x_n| \leq |x_1| + |x_2 + ... +x_n |\leq ... \leq |x_1 | +...+|x_n |$. Thanks from topsquark and woo Last edited by idontknow; November 6th, 2019 at 10:57 AM. November 6th, 2019, 11:53 AM   #3
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Quote:
 Originally Posted by woo How to show that $\displaystyle \lim_{M\to\infty}\left|\sum_{n=1}^{M}a_n\right| = \left|\sum_{n=1}^{\infty}a_n\right|$? Thank you.
Absolute value is a continuous function which means it commutes with limits. Specifically, this implies the following equality:
$\lim_{M\to\infty}\left|\sum_{n=1}^{M}a_n\right| = \left|\lim_{M\to\infty}\sum_{n=1}^{M}a_n \right|$

Last edited by skipjack; November 6th, 2019 at 04:26 PM. Tags absolutely, approach, convergent, infinity, partial, series, sum Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post Dattier Real Analysis 14 June 27th, 2017 06:55 AM Sofica Real Analysis 0 November 3rd, 2014 04:22 PM Dmath Calculus 2 June 4th, 2014 07:22 AM nappysnake Real Analysis 1 December 18th, 2011 07:49 PM The Chaz Real Analysis 11 February 7th, 2011 05:52 AM

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