- #1

k3k3

- 78

- 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.