February 17th, 2018, 09:21 AM  #1 
Member Joined: Dec 2017 From: Tel Aviv Posts: 43 Thanks: 3  Convergence Last edited by greg1313; February 17th, 2018 at 11:35 AM. 
February 17th, 2018, 09:33 AM  #2 
Math Team Joined: Jul 2011 From: Texas Posts: 2,722 Thanks: 1375  as posted, this expression represents a sequence, not a series. The sequence converges ... $\displaystyle \lim_{n \to \infty} \bigg[1 + \dfrac{(1)^n}{n}\bigg] = 1$ The series diverges ... $\displaystyle \sum_{n=1}^\infty \bigg[1 + \dfrac{(1)^n}{n}\bigg]$ ... if the individual terms of a series (in other words, the terms of the series' underlying sequence) do not converge to zero, then the series must diverge. This is the nth term test for series divergence. 
February 17th, 2018, 09:39 AM  #3 
Member Joined: Dec 2017 From: Tel Aviv Posts: 43 Thanks: 3 
Sorry, I not understood. What the difference between series to sequence? Can you write in some word what the definition of these terms? Thank... 
February 17th, 2018, 10:31 AM  #4  
Senior Member Joined: May 2016 From: USA Posts: 917 Thanks: 368  Quote:
Basically, a sequence defines an enumerated group of numbers. Your formula may define the infinite sequence $x_1 = 1 + \dfrac{\ 1^1}{1} = 0.$ $x_2 = 1 + \dfrac{\ 1^2}{2} = \dfrac{3}{2}.$ $x_3 = 1 + \dfrac{\ 1^3}{3} = \dfrac{2}{3}.$ $x_4 = 1 + \dfrac{\ 1^4}{4} = \dfrac{5}{4}.$ https://en.wikipedia.org/wiki/Series_(mathematics) An infinite series is the sum of an infinite sequence.  
February 17th, 2018, 10:33 AM  #5 
Member Joined: Dec 2017 From: Tel Aviv Posts: 43 Thanks: 3 
Thanks, i don't know if your definitions are suffice to know the mattrials. But, Now I "think" that I know the different. 
February 17th, 2018, 10:34 AM  #6 
Senior Member Joined: Sep 2016 From: USA Posts: 306 Thanks: 157 Math Focus: Dynamical systems, analytic function theory, numerics 
How can you ask about convergence of a sequence of series if you don't even know what they are? I think you need to spend more time reading and understanding and less time trying to finish homework as quickly as possible.

February 17th, 2018, 10:42 AM  #7 
Member Joined: Dec 2017 From: Tel Aviv Posts: 43 Thanks: 3 
I should listen to you. Nice. I will take a break from math in the next days. 