My Math Forum  

Go Back   My Math Forum > College Math Forum > Calculus

Calculus Calculus Math Forum


Thanks Tree6Thanks
  • 2 Post By idontknow
  • 4 Post By SDK
Reply
 
LinkBack Thread Tools Display Modes
November 6th, 2019, 10:19 AM   #1
woo
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.
woo is offline  
 
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.
idontknow is offline  
November 6th, 2019, 11:53 AM   #3
SDK
Senior Member
 
Joined: Sep 2016
From: USA

Posts: 683
Thanks: 456

Math Focus: Dynamical systems, analytic function theory, numerics
Quote:
Originally Posted by woo View Post
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:
\[
\lim_{M\to\infty}\left|\sum_{n=1}^{M}a_n\right| = \left|\lim_{M\to\infty}\sum_{n=1}^{M}a_n \right|
\]
Thanks from topsquark, woo, romsek and 1 others

Last edited by skipjack; November 6th, 2019 at 04:26 PM.
SDK is online now  
Reply

  My Math Forum > College Math Forum > Calculus

Tags
absolutely, approach, convergent, infinity, partial, series, sum



Thread Tools
Display Modes


Similar Threads
Thread Thread Starter Forum Replies Last Post
Convergent series : cos(k^2)/k Dattier Real Analysis 14 June 27th, 2017 06:55 AM
Fourier series absolutely convergence Sofica Real Analysis 0 November 3rd, 2014 04:22 PM
Convergent Series Dmath Calculus 2 June 4th, 2014 07:22 AM
Is this series convergent? nappysnake Real Analysis 1 December 18th, 2011 07:49 PM
Convergent series -> series of geometric means converges The Chaz Real Analysis 11 February 7th, 2011 05:52 AM





Copyright © 2019 My Math Forum. All rights reserved.