Series convergence

[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 |an+1|/|an| 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 (an)1/n goes to a number less than 1, then the series converges, etc.

With a power series &Sigma;anxn, the ratio becomes |a[sub]n+1]|/a[sub]n[/sup]| |x|< 1. As long as |a[sub]n+1]|/a[sub]n[/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.