Series Convergence: Region in xy-Plane & Tests to Use

[yxgao]

Let D be the region in the xy-plane in which the series
Sum k=1 to k= infinity (x+2y)^k /k converges. Then the interior of D is: The open region between two parallel lines.

Can someone explain why this is true? You don't need to work out a full blown solution.

What convergence tests are there for series and when should each be used?

Thanks!

 

[HallsofIvy]

This is a power series.

In general, either the "ratio test" or "root test" work nicely with powers series.

The "ratio test" (which I am sure you covered when you first learned about infinite series) says that if the fraction |a_n+1|/|a_n| converges to a number less than 1 then the series converges. If to a number larger than 1, diverges. (if to 1, you need more information.)

The "root test" often works when the ratio test fails (gives a result when the ratio goes to 1) but is usually harder to apply.
If (a_n)^1/n goes to a number less than 1, then the series converges, etc.

With a power series &Sigma;a_nxⁿ, the ratio becomes |a<sub>n+1]|/an[/sup]| |x|< 1. As long as |an+1]|/an[/sup]| itself converges (to A, say), the series will converge (absolutely) as long |x|< A, diverge for |x|> A, may or may not converge at |x|= A. A is the "radius of convergence".

For something like (x+2y)^k /k, the fact that you have two variables x and y is irrelevant. For fixed x,y they are just constants. We still require that |k+1/k| |x+2y| must have a limit less that 1. Since |k+1/k| has limit 1, the series converges as long as |x+ 2y|<1 which means -1< x+ 2y< 1. That is the region bounded by the parallel straight lines x+ 2y= -1 and x+ 2y= 1.</sub>

 

[M98Ranger]

The statement that the interior of D is the open region between two parallel lines can be explained by considering the convergence criteria for the given series. The series can be rewritten as (x^k / k) + (2y)^k / k. Now, for the series to converge, the terms in the series must approach zero as k approaches infinity. This means that both (x^k / k) and (2y)^k / k must approach zero.

For (x^k / k) to approach zero, x must be less than 1 in absolute value. Similarly, for (2y)^k / k to approach zero, 2y must be less than 1 in absolute value. This means that the values of x and y must lie within certain ranges for the series to converge.

Now, let us consider the equation x+2y=1. This equation represents a line in the xy-plane. For any point (x,y) that satisfies this equation, the series will not converge as either (x^k / k) or (2y)^k / k will not approach zero. Therefore, the interior of D, where the series converges, must lie outside this line.

Similarly, if we consider the equation x+2y=-1, we can see that the interior of D must also lie outside this line. Therefore, the interior of D must be the open region between these two parallel lines, as any point within this region will satisfy the conditions for the series to converge.

As for the convergence tests for series, there are several tests that can be used depending on the nature of the series. Some commonly used tests include the ratio test, the root test, the comparison test, and the integral test.

The ratio test is used to determine the convergence or divergence of a series by comparing the ratio of successive terms in the series to a limit. The root test is similar to the ratio test, but it compares the nth root of the absolute value of the terms to a limit.

The comparison test is used to compare a given series to a known convergent or divergent series to determine its convergence. The integral test is used to determine the convergence of a series by comparing it to a related improper integral.

Each of these tests has its own conditions for use and can be applied to different types of series. It is important to understand the conditions and limitations of each test in order to determine which one is most