Region of Convergence for Series in x+2y: Description and Solution

In summary, the region in the xy plane in which the given series converges is a strip with a radius of convergence of 1, where x+2y is less than 1 and greater than or equal to -1. The series will not converge outside of this strip.
  • #1
ehrenfest
2,020
1
[SOLVED] radius of convergence

Homework Statement


Let D be th region in the xy plane in which the series
[tex]\sum_{k=1}^{\infty}\frac{(x+2y)^k}{k}[/tex]
converges. Describe D.

Homework Equations


The Attempt at a Solution


By the ratio test, we find the radius of converge of the series in x+ 2y to be 1. So, the series will converge when |x+2y| < 1. This region is a rectangle.

What is wrong with this?
 
Physics news on Phys.org
  • #2
Be careful with the ratio test because it only tests for absolute convergence. However you are close. It should be [tex]x+2y < 1[/tex] and [tex]x+2y \geq -1[/tex]. This is not a rectangle. It's a "strip" through the plane.
 
  • #3
I am an idiot.
 

What is the radius of convergence?

The radius of convergence is a mathematical concept that refers to the distance from the center of a power series to the point where it converges. It is denoted by the letter "R" and is typically measured in terms of the variable x in the power series.

How do you calculate the radius of convergence?

The radius of convergence can be calculated using the ratio test, which involves taking the limit of the absolute value of the ratio of consecutive terms in the power series. If this limit is less than 1, then the series will converge. The distance from the center of the series to the point where it converges is the radius of convergence.

What is the significance of the radius of convergence?

The radius of convergence is an important concept in calculus and analysis as it determines the interval of values for which a power series is valid. It also helps to determine whether a function can be represented by a power series, and if so, what is the interval of convergence.

What happens if the limit in the ratio test is equal to 1?

If the limit in the ratio test is equal to 1, then the series may or may not converge. Further analysis is needed to determine the convergence or divergence of the series, such as using the root test or other convergence tests.

Can the radius of convergence be negative?

No, the radius of convergence is always a positive value. It represents a distance and therefore cannot be negative. However, the interval of convergence can be negative if the series is centered at a negative value.

Similar threads

Interval of convergence of power series from ratio test
    • Calculus and Beyond Homework Help
Replies
2
Views
189
For what values of ##\beta## does the series converge?
    • Calculus and Beyond Homework Help
Replies
2
Views
738
On the ratio test for power series
    • Calculus and Beyond Homework Help
Replies
2
Views
711
How to show that these sequences are summable?
    • Calculus and Beyond Homework Help
Replies
1
Views
259
On pointwise convergence of Fourier series
    • Calculus and Beyond Homework Help
Replies
3
Views
418
Set of convergence of a Power series
    • Calculus and Beyond Homework Help
Replies
4
Views
1K
Addition of power series and radius of convergence
    • Calculus and Beyond Homework Help
Replies
14
Views
1K
Possible Values of x for Convergence of Power Series
    • Calculus and Beyond Homework Help
Replies
1
Views
917
Checking series for convergence
    • Calculus and Beyond Homework Help
Replies
5
Views
489
Intervals of Convergence- Power Series
    • Calculus and Beyond Homework Help
Replies
11
Views
2K

Hot Threads

Recent Insights

Back
Top