Showing the sum of convergent and divergent sequence is divergent

In summary: If {x_n} converges to x, but {y_n} diverges, the {x_n} terms are what keep the sequence from diverging. So it is safe to put both sequences on top of each other.
  • #1
k3k3
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0

Homework Statement



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?

Homework 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.

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.
 
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  • #2
k3k3 said:
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.
 
  • #3
micromass said:
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.
 
  • #4
k3k3 said:
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??
 
  • #5
micromass said:
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.
 
  • #6
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.
 
  • #7
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.
 

FAQ: Showing the sum of convergent and divergent sequence is divergent

What is a convergent sequence?

A convergent sequence is a sequence of numbers that approaches a specific limit as the number of terms increases. In other words, as the sequence continues, the terms get closer and closer to a single value.

What is a divergent sequence?

A divergent sequence is a sequence of numbers that does not approach a specific limit as the number of terms increases. In other words, as the sequence continues, the terms do not get closer to a single value and may either increase or decrease without end.

How can you show that the sum of a convergent and divergent sequence is divergent?

This can be shown by using the definition of a convergent sequence. If a sequence is convergent, then the sum of its terms will approach a specific limit. However, since a divergent sequence does not have a limit, the sum of a convergent and divergent sequence will also not have a limit and thus, will be divergent.

Can a convergent and divergent sequence have the same limit?

No, a convergent and divergent sequence cannot have the same limit. This is because a convergent sequence approaches a specific limit while a divergent sequence does not have a limit.

Is it possible for a sequence to be both convergent and divergent?

No, a sequence cannot be both convergent and divergent. It can only be one or the other. A sequence can only have one limit, and if it approaches a specific value as the number of terms increases, it is convergent. If it does not approach a specific value, it is divergent.

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