Prove that the sequence converges to 0 (2)

MHB
Thread starter alexmahone
Start date Jan 27, 2012
Tags

Sequence

In summary, when a sequence converges to 0, it means that the terms in the sequence approach 0 as the number of terms increases. To prove this, we use the definition of convergence and show that the terms get closer and closer to 0. This is important because it helps us understand the behavior of the sequence and determine its limit. A sequence can converge to 0 from both positive and negative values. Some common techniques used to prove this include the squeeze theorem, the ratio test, and the root test.

Jan 27, 2012

alexmahone

e_{n+1} = (e_n-2)/(e_n+4)

Prove that {e_n} converges to 0 if

(a) e_0 > -1

(b) -2 < e_0 < -1

PS: I haven't learned things like sup and inf yet, so please don't use them.

Physics news on Phys.org

Jan 27, 2012

ThePerfectHacker

Look at the other question you posted. The idea is very similar.
Also it would help if you post your work so we can help you with your steps.

Jan 27, 2012

alexmahone

I got it. Thanks!

Related to Prove that the sequence converges to 0 (2)

1. What is meant by "converges to 0" in a sequence?

When a sequence converges to 0, it means that the terms in the sequence approach 0 as the number of terms increases. In other words, the values in the sequence get closer and closer to 0 as the sequence progresses.

2. How do you prove that a sequence converges to 0?

To prove that a sequence converges to 0, we use the definition of convergence. This means showing that the terms in the sequence get closer and closer to 0 as the index of the sequence increases, and that the difference between the terms and 0 can be made arbitrarily small.

3. What is the importance of proving that a sequence converges to 0?

Proving that a sequence converges to 0 is important because it allows us to understand the behavior of the sequence and make predictions about its future values. It also helps us determine the limit of the sequence, which is a fundamental concept in calculus and other mathematical fields.

4. Can a sequence converge to 0 from both positive and negative values?

Yes, a sequence can converge to 0 from both positive and negative values. This is known as a two-sided limit. It means that the terms in the sequence get closer and closer to 0 from both sides as the index of the sequence increases.

5. What are some common techniques used to prove that a sequence converges to 0?

Some common techniques used to prove that a sequence converges to 0 include the squeeze theorem, the ratio test, and the root test. These techniques involve manipulating the terms in the sequence and using the definition of convergence to show that the terms get closer and closer to 0.