Convergence/Divergence of an Infinite Series

In summary: This is a telescoping series, and the limit as ##N \to \infty## is ##\frac{1}{2} \times \frac{1}{2} = \frac{1}{4} > 0##. We conclude that the series diverges.In summary, the series in question is divergent and can be proven using the limit comparison test or by comparing it to a known divergent series, such as the harmonic series.
  • #1
Alex Myhill
5
0

Homework Statement


To Determine Whether the series seen below is convergent or divergent.

Homework Equations


∑(n/((n+1)(n+2))) From n=1 to infinity.

The Attempt at a Solution


Tried to use the comparison test as the bottom is n^2 + 3n + 2, comparing to 1/n. However, this does not work as the first series is "smaller" than the harmonic series and thus it can't be proved to be divergent. I also tried to separate it into partial fractions but I don't know how to prove it is divergent from there. Thanks a lot for any help.
 
Physics news on Phys.org
  • #2
Alex Myhill said:

Homework Statement


To Determine Whether the series seen below is convergent or divergent.

Homework Equations


∑(n/((n+1)(n+2))) From n=1 to infinity.

The Attempt at a Solution


Tried to use the comparison test as the bottom is n^2 + 3n + 2, comparing to 1/n. However, this does not work as the first series is "smaller" than the harmonic series and thus it can't be proved to be divergent. I also tried to separate it into partial fractions but I don't know how to prove it is divergent from there. Thanks a lot for any help.
Is it still smaller if you start the harmonic series at six? And does this make the harmonic series convergent?
 
  • #3
Alex Myhill said:

Homework Statement


To Determine Whether the series seen below is convergent or divergent.

Homework Equations


∑(n/((n+1)(n+2))) From n=1 to infinity.

The Attempt at a Solution


Tried to use the comparison test as the bottom is n^2 + 3n + 2, comparing to 1/n. However, this does not work as the first series is "smaller" than the harmonic series and thus it can't be proved to be divergent. I also tried to separate it into partial fractions but I don't know how to prove it is divergent from there. Thanks a lot for any help.
Have you seen the limit comparison test?

Also, this is definitely not a precalculus problem, so I have moved it to the calculus section.
 
  • #4
Hi,
Thanks for the response. So to prove using comparison just start harmonic series at six. Makes sense, I hadn't thought of that.
Referring to Mark44's response, no, I haven't heard of the limit comparison test, however, I will look it up.
Thanks heaps for your time and input.
 
  • #5
Alex Myhill said:
Hi,
Thanks for the response. So to prove using comparison just start harmonic series at six. Makes sense, I hadn't thought of that.
Referring to Mark44's response, no, I haven't heard of the limit comparison test, however, I will look it up.
Thanks heaps for your time and input.

I don't see how starting the harmonic series at six solves the problem, unless you also do some other things at the same time.

To me, the simplest solution is to note that for ##n \geq 1## we have ##n/(n+1) \geq 1/2##, so that
$$t_n \equiv \frac{n}{(n+1)(n+2)} \geq \frac{1}{2} \frac{1}{n+2}.$$
That means
$$\sum_{n=1}^N t_n \geq \sum_{n=1}^N \frac{1}{2} \frac{1}{n+2} = \frac{1}{2} \left[\frac{1}{3}+ \frac{1}{4} + \cdots + \frac{1}{N+2} \right] $$
 

1. What is the definition of convergence and divergence of an infinite series?

The convergence of an infinite series is when the sum of its terms approaches a finite value as the number of terms increases. In contrast, the divergence of an infinite series is when the sum of its terms does not approach a finite value as the number of terms increases.

2. How do you determine if an infinite series is convergent or divergent?

There are several tests that can be used to determine the convergence or divergence of an infinite series, such as the comparison test, ratio test, and integral test. These tests involve comparing the given series to a known convergent or divergent series and using algebraic and/or calculus techniques to evaluate the limit of the series.

3. What is the importance of understanding convergence and divergence of infinite series?

Understanding convergence and divergence of infinite series is important in many areas of mathematics and science, as it allows us to determine the behavior of a series and make predictions about its sum. In addition, it is a fundamental concept in calculus and is used in many applications, such as in the analysis of functions and the study of sequences and series.

4. Can an infinite series be both convergent and divergent?

No, an infinite series cannot be both convergent and divergent. It must be one or the other. However, a series can be conditionally convergent, which means that it is convergent but the sum of its terms can change depending on the order in which they are added, or absolutely convergent, which means that it is convergent regardless of the order in which its terms are added.

5. What is the role of the limit in determining convergence and divergence of an infinite series?

The limit plays a crucial role in determining the convergence or divergence of an infinite series. In order for a series to be convergent, the limit of its terms must approach a finite value as the number of terms increases. If the limit does not exist or approaches infinity, the series is divergent. The limit also helps us evaluate the convergence or divergence of a series using various tests.

Similar threads

Interval of convergence of power series from ratio test
    • Calculus and Beyond Homework Help
Replies
2
Views
178
How to show that these sequences are summable?
    • Calculus and Beyond Homework Help
Replies
1
Views
249
On the ratio test for power series
    • Calculus and Beyond Homework Help
Replies
2
Views
708
Am I allowed to solve this series this way?
    • Calculus and Beyond Homework Help
Replies
29
Views
1K
Doubt regarding the series ##\sum [\sqrt{n+1} - \sqrt{n}\,]##
    • Calculus and Beyond Homework Help
Replies
17
Views
1K
Learning to use the Cauchy criterion for infinite series
    • Calculus and Beyond Homework Help
Replies
7
Views
1K
Studying the convergence of a series with an arctangent of a partial sum
    • Calculus and Beyond Homework Help
Replies
7
Views
951
A problem with the convergence of a series
    • Calculus and Beyond Homework Help
Replies
5
Views
985
Does the Alternating Series Test show convergence for this series?
    • Calculus and Beyond Homework Help
Replies
14
Views
1K
Finding the Limit of a Series: Exploring Different Patterns and Approaches
    • Calculus and Beyond Homework Help
Replies
2
Views
847

Hot Threads

Recent Insights

Back
Top