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April 4th, 2018, 10:21 AM  #1 
Newbie Joined: Apr 2018 From: India Posts: 4 Thanks: 0  Infinite convergent series.
How do I find the limiting value of any convergent series ? I mean, how do I approach it? ("steps", maybe?) ex : (1^2)+(2^2)+(3^2)+.... = pi^2/6 I know the solution of this problem but how on earth did anyone (Euler) come up with this ? . . . . . ( I'm trying to solve this (1^3)+(2^3)+(3^3)+..... = ? ) 
April 4th, 2018, 10:30 AM  #2 
Senior Member Joined: Sep 2015 From: USA Posts: 2,647 Thanks: 1476  How Euler found the sum of inverse squares I don't believe there is a general strategy to your problem. You can try to match up the coefficients with a Taylor series. A handful of series, such as the geometric series, have known results. 
April 4th, 2018, 10:45 AM  #3 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,700 Thanks: 2682 Math Focus: Mainly analysis and algebra  
April 4th, 2018, 01:40 PM  #4 
Math Team Joined: May 2013 From: The Astral plane Posts: 2,345 Thanks: 986 Math Focus: Wibbly wobbly timeywimey stuff. 
That was awesome. Thanks to romsek and v8archie for posting them. Dan 
April 4th, 2018, 03:35 PM  #5 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,700 Thanks: 2682 Math Focus: Mainly analysis and algebra 
Just remembered this mildly awesome one too, although it has less to do with Euler. 

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convergent, infinite, series 
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