Register to reply

Showing the sum of convergent and divergent sequence is divergent

Share this thread:
k3k3
#1
Mar31-12, 11:34 AM
P: 78
1. The problem statement, all variables and given/known data

Show that the sum of a convrgent sequence and a divergent sequence must be a divergent sequence. What can you say about the sum of two divergent sequences?

2. Relevant equations
A theorem in the book states:
Let {a_n} converge to a and {b_n} converge to b, then the sequence {a_n+b_n} converges to a+b

Definition of convergent:
A sequence {a_n} converges to a number L if for each epsilon > 0 there exists a positive integer N such that |a_n-L| < epsilon for all n ≥ N. The number L is called the limit of the sequence. The sequence {a_n} converges iuf there exists a number L such that {a_n} converges to L. The sequence {x_n} diverges if it does not converge.

3. The attempt at a solution
I think I made this too simple and overlooked something... Here is my attempt.

Let {x_n} be a sequence that converges to x.
Let {y_n} be a divergent sequence.

Suppose {x_n+y_n} is a convergent sequence.
By the theorem in the book, this implies that {y_n} converges, but {y_n} diverges.
Hence, the sum must be divergent.

What can you say about the sum of two divergent sequences? Their sum is divergent.
Phys.Org News Partner Science news on Phys.org
Flapping baby birds give clues to origin of flight
Prions can trigger 'stuck' wine fermentations, researchers find
Socially-assistive robots help kids with autism learn by providing personalized prompts
micromass
#2
Mar31-12, 11:37 AM
Mentor
micromass's Avatar
P: 18,291
Quote Quote by k3k3 View Post
By the theorem in the book, this implies that {y_n} converges
How does that follow from the theorem in the book?? What do you take as a_n and b_n??

What can you say about the sum of two divergent sequences? Their sum is divergent.
Not necessarily.
k3k3
#3
Mar31-12, 11:42 AM
P: 78
Quote Quote by micromass View Post
How does that follow from the theorem in the book?? What do you take as a_n and b_n??
If we assume that the sum of the convergent sequence and divergent sequence is convergent, and use that the theorem the book states, both sequences must be convergent. There should be some number that {y_n} converges to but there isn't, so it can't be.

micromass
#4
Mar31-12, 11:43 AM
Mentor
micromass's Avatar
P: 18,291
Showing the sum of convergent and divergent sequence is divergent

Quote Quote by k3k3 View Post
If we assume that the sum of the convergent sequence and divergent sequence is convergent, and use that the theorem the book states, both sequences must be convergent. There should be some number that {y_n} converges to but there isn't, so it can't be.
Yes, but what do you take as a_n and b_n??
k3k3
#5
Mar31-12, 11:44 AM
P: 78
Quote Quote by micromass View Post
Yes, but what do you take as a_n and b_n??
{x_n} is one of them and I was wanting to show that {y_n} is neither.
DrewD
#6
Mar31-12, 11:56 AM
P: 446
Let {a_n} converge to a and {b_n} converge to b, then the sequence {a_n+b_n} converges to a+b
This is a one way implication. It does not imply that if the sum is convergent then the two initial series are convergent. Note though, that the difference between two convergent series is convergent.
k3k3
#7
Mar31-12, 12:12 PM
P: 78
Would it be safe to put both sequences on top of each other? Since they are sequences, they are in 1-1 correspondence with the natural numbers. So {x_n+y_n}=x_1,y_1,...,x_n,y_n,...

If {x_n} converges to x, but {y_n} converges to nothing, the {y_n} terms are what break the sequence from converging.


Register to reply

Related Discussions
Convergent or Divergent Calculus & Beyond Homework 7
Showing a sequence is divergent Calculus 5
Convergent or divergent? Calculus & Beyond Homework 11
Limit of a convergent series and a divergent sequence Calculus & Beyond Homework 0
Convergent and Divergent problem Calculus & Beyond Homework 2