Telescoping Sum Convergence: Explained and Solved with Examples | Homework Help

  • Thread starter Fuzedmind
  • Start date
  • Tags
    Issues Sum
In summary, the problem asks for the sum of a series expressed as a telescoping sum and to determine its convergence or divergence. The student struggles with understanding how to solve it, particularly with the partial fractions. After some guidance, the student is able to find the correct solution.
  • #1
Fuzedmind
9
0

Homework Statement



The problem asks me to express the sum of the series as a telescoping sum, then find whether it is convergent or divergent. Ok, I get that and how it works and all, but the examples they give in the book are stupid and i on spring break this week so no office hours for professors.

Homework Equations


Here it is:

2/(n^2 + 4n + 3)

I know, easy, but I don't get how to do it...the easy ones stump me.

The Attempt at a Solution



I rewrote it like this:
(1/2)(2/n+3) - 2/n+1)

But the terms do not cancel when I do this. Plus it is an even question so I do not know the solution.
 
Physics news on Phys.org
  • #2
Fuzedmind said:
..but the examples they give in the book are stupid and i on spring break this week

I can tell :rofl:

You didn't do the partial fractions right; This is apparent by plugging in n=0 to the original and what you got: [tex]\frac{2}{n^2+4n+3}=\frac{1}{n+1}-\frac{1}{n+3}[/tex]. These terms WILL cancel at some point. Write out the first 5 or so terms of the series and you will see this.
 
  • #3
Well I did that, and they started cancelling, and I got

(1 - 1/3) + (1/2 - 1/4) + (1/3 - 1/5) + (1/4 - 1/6) + (1/5 - 1/7)

I canceled the 1/3, 1/4, and the 1/5 out, but where do I go from there?

Sorry I am kind of retarted
 
  • #4
Now write a few more terms and cancel the 1/6 and 1/7. What terms don't cancel? I kind of have faith that you aren't THAT retarded.
 
  • #5
What you want to do is make sure that, for each "-1/(n+1)", there exist an m so that its "1/(m+3)" cancels it. That is, given an integer n, what m will make 1/(m+3)= 1/(n+1)?
 

What are telescoping sum issues?

Telescoping sum issues refer to the challenges that arise when trying to solve a mathematical problem that involves an infinite series or sum, where each term in the series depends on the previous one.

What is the difference between telescoping and non-telescoping sums?

A telescoping sum is one in which most of the terms cancel out, leaving only a few terms at the beginning and end. In contrast, a non-telescoping sum is one in which all the terms are needed to calculate the final result.

How do you simplify a telescoping sum?

To simplify a telescoping sum, you need to identify the pattern in the terms and use it to reduce the sum to a more manageable form. This often involves grouping terms or factoring out common factors.

What are some common strategies for solving telescoping sums?

One common strategy is to look for a way to rewrite the terms in the sum in a simpler form, such as using partial fractions or logarithms. Another strategy is to use a telescoping product, where each term depends on the previous one but also has a constant factor that simplifies the sum.

Why are telescoping sums important in mathematics?

Telescoping sums are important because they allow us to solve complex mathematical problems by reducing them to simpler forms. They also have many real-world applications, such as in engineering, physics, and finance.

Similar threads

  • Calculus and Beyond Homework Help
Replies
7
Views
919
  • Calculus and Beyond Homework Help
Replies
7
Views
1K
  • Calculus and Beyond Homework Help
Replies
23
Views
2K
  • Calculus and Beyond Homework Help
Replies
3
Views
1K
  • Calculus and Beyond Homework Help
Replies
5
Views
820
  • Calculus and Beyond Homework Help
Replies
2
Views
4K
  • Calculus and Beyond Homework Help
Replies
13
Views
995
  • Calculus and Beyond Homework Help
Replies
6
Views
998
  • Calculus and Beyond Homework Help
Replies
6
Views
1K
  • Calculus and Beyond Homework Help
Replies
2
Views
2K
Back
Top