Convergence of Infinite Series

In summary, determining whether the infinite series \sum_{n=1}^{\infty } \frac{1}{n^{1+1/n}} converges is a complex problem. The integral test is not applicable and the comparison tests and ratio test are inconclusive. However, the limit of n^(1/n) suggests that the series converges. By comparing it to a series with a maximum value, it can be shown that the original series also has a maximum value and therefore converges. The limit of n^(1/n) is 1 and for all n in the sum, it is also less than or equal to 3^(1/3). This can be proven by showing that x^(1/x) decreases
  • #1
aostraff
48
0
I need some help on determining whether this infinite series converges (taken from Spivak for those curious):

[tex]\sum_{n=1}^{\infty } \frac{1}{n^{1+1/n}}[/tex]

I would think the integral test would be most appropriate but it doesn't seem to work (because the integral seems hard). The obvious comparison tests are inconclusive and a ratio test seems inconclusive as well. I'm guessing the best idea right now would be to think of comparisons. Thanks.
 
Last edited:
Physics news on Phys.org
  • #2
Your denominator is n^(1+1/n)=n*n^(1/n). Can you figure out the limit of n^(1/n)? Does that suggest a comparison?
 
  • #3
Yeah it does. So it converges. Thanks.
 
  • #4
aostraff said:
Yeah it does. So it converges. Thanks.

No it doesn't! What are you going to compare with? Are you sure it converges?!
 
  • #5
Uh oh. I'm back to square one. I thought I had it but I was being reckless. I'm not quite sure how the existence of a limit for n^(1/n) helps me.
 
  • #6
What IS the limit? The existence of a limit implies n^(1/n) has a maximum value. Can you find an M such that n^(1/n)<M. Now what series to compare with?
 
  • #7
so n^(1/n) in limit goes to 1...

think about what type of sum this leaves you with as we go towards the limit

it is also< 2 for all n in the sum (actually <=3^(1/3) can show x^(1/x) decreases monotonically for lnx>1)
 
Last edited:

1. What is the definition of convergence of infinite series?

Convergence of infinite series is a mathematical concept that refers to the idea that an infinite sum of terms can have a finite limit or value. In other words, as the number of terms in the series increases, the sum can approach a specific value rather than continuing to increase indefinitely.

2. How do you determine if an infinite series converges or diverges?

There are several tests that can be used to determine the convergence or divergence of an infinite series, including the comparison test, the ratio test, and the integral test. These tests involve evaluating the behavior of the terms in the series and comparing them to known convergent or divergent series.

3. What is the difference between absolute convergence and conditional convergence?

Absolute convergence refers to the convergence of an infinite series when all terms are positive or all terms are negative. Conditional convergence, on the other hand, occurs when the series converges but would diverge if the terms were rearranged. In other words, the order of the terms affects the convergence of the series.

4. Can an infinite series converge to more than one value?

No, an infinite series can only converge to one value. This is because the definition of convergence states that as the number of terms in the series increases, the sum approaches a specific value. Therefore, the series can only have one limit or value that it converges to.

5. What are some real-world applications of convergence of infinite series?

Convergence of infinite series has many practical applications in fields such as physics, engineering, and finance. For example, it is used to model the behavior of electrical circuits, analyze the stability of structures, and calculate the present value of future cash flows in financial investments. It is also used in computer algorithms and data analysis to approximate complex functions and solve optimization problems.

Similar threads

How to show that these sequences are summable?
    • Calculus and Beyond Homework Help
Replies
1
Views
259
Interval of convergence of power series from ratio test
    • Calculus and Beyond Homework Help
Replies
2
Views
189
On the ratio test for power series
    • Calculus and Beyond Homework Help
Replies
2
Views
711
For what values of ##\beta## does the series converge?
    • Calculus and Beyond Homework Help
Replies
2
Views
738
On pointwise convergence of Fourier series
    • Calculus and Beyond Homework Help
Replies
3
Views
418
Series Convergence: What Can the Nth Term Test Tell Us?
    • Calculus and Beyond Homework Help
Replies
6
Views
1K
Using comparison tests and limit comparison test
    • Calculus and Beyond Homework Help
Replies
1
Views
1K
Prove that this series converges uniformly on R
    • Calculus and Beyond Homework Help
Replies
4
Views
311
On the definition of radius of convergence; a small supremum technicality
    • Calculus and Beyond Homework Help
Replies
7
Views
709
Study the convergence and absolute convergence of the following series
    • Calculus and Beyond Homework Help
Replies
5
Views
1K

Hot Threads

Recent Insights

Back
Top