Convergence of a series

In summary, the proof shows that if {an} is bounded and {bn} converges to 0, then {anbn} also converges to 0. However, if {an} is not bounded, then {anbn} may not necessarily converge. An example is given where {an} does not converge to 0, and thus {anbn} does not converge to 0.
  • #1
25
0

Homework Statement


Consider sequences {an} and {bn}, where sequence {bn} converges to 0.
Is it true that sequence {anbn} converges to 0?



The Attempt at a Solution



Proof.
First I assumed (an) is bounded, and so there exists M > 0 such that |an| < M for all n 2
{1, 2, 3, . . .}. Moreover, since lim bn = 0, there exists N such that, for all n≥ N,
|bn − 0| <ε /M
or, equivalently, |bn| < ε/M.
Therefore, whenever n ≥ N, we have that
|anbn − 0| = |anbn| = |an||bn| ≤M|bn| < M(ε/M)=ε

Since the choice of  ε> 0 was arbitrary, this implies that lim(anbn) = 0.

However, I know that {anbn} can also not converge if {an} is not bounded. Can someone help with how to go about that part of hte proof?
Thank you!
 
Physics news on Phys.org
  • #2
It obviously doesn't work when an is not bounded. I am understanding correctly that an does not converge to 0, right?
if so, just consider bn=1/n and an=n^2. an*bn=n does not convergen. Bam!
 
  • #3
There isn't given any information on if {an} converges to 0 or not. Thanks
 

What is the definition of convergence of a series?

The convergence of a series is the property of a series to approach a finite limit or value as the number of terms in the series increases. In other words, the series can be said to "converge" towards a specific value.

How is convergence of a series different from divergence?

Convergence and divergence are two opposite properties of a series. While convergence means that the series approaches a finite limit, divergence means that the series does not have a finite limit and instead grows towards infinity.

What is the importance of convergence of a series in mathematics?

The concept of convergence of a series is crucial in many branches of mathematics, including calculus, analysis, and number theory. It allows us to determine the behavior and properties of infinite series and to make mathematical calculations and predictions based on these properties.

What are some common tests for determining the convergence of a series?

Some common tests for determining the convergence of a series include the ratio test, the comparison test, the integral test, and the root test. These tests help us determine whether a series converges or diverges based on certain criteria.

What are some real-life applications of convergence of a series?

The concept of convergence of a series has practical applications in various fields, such as engineering, physics, and finance. For example, engineers use series convergence to model and predict the behavior of electrical circuits, while physicists use it to calculate the behavior of particles in a system. In finance, the concept is used in the calculation of compound interest and in the evaluation of risk management strategies.

Suggested for: Convergence of a series

Replies
5
Views
370
Replies
3
Views
253
Replies
2
Views
459
Replies
10
Views
839
Replies
8
Views
940
Replies
11
Views
1K
Replies
23
Views
2K
Replies
6
Views
949
Replies
5
Views
1K
Replies
5
Views
902
Back
Top