Interval of convergence of power series from ratio test

  • #1
zenterix
693
83
Homework Statement
While reading a chapter from Simmons' "Differential Equations with Applications and Historical Notes" that reviews power series, I did not really understand a paragraph that tried to explain how to obtain the radius of convergence of a power series using the ratio test.
Relevant Equations
Consider the following power series in ##x##

$$\sum\limits_{n=0}^\infty a_nx^n$$
If ##a_n\neq 0## for all ##n##, consider the limit

$$\lim\limits_{n\to\infty} \left | \frac{a_{n+1}x^{n+1}}{a_nx^n} \right | = \lim\limits_{n\to\infty}\left |\frac{a_{n+1}}{a_n}\right | |x|=L$$

The ratio test asserts that this series converges if ##L<1## and diverges if ##L>1##.

These considerations yield the formula

$$R=\lim\limits_{n\to\infty}\left |\frac{a_{n+1}}{a_n}\right |\tag{1}$$

if this limit exists (we put ##R=\infty## if ##|a_n/a_{n+1}|\to\infty##).

Regardless of whether this formula can be used or not, it is known that ##R## always exists; and if ##R## is finite and nonzero, then it determines an interval of convergence ##-R<x<R## such that inside the interval the series converges and outside the interval it diverges.

I'm a bit confused by this snippet.

Why does (1) determine the radius of convergence?
 
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  • #2
I guess it is because of the following

$$\lim\limits_{n\to\infty}\left |\frac{a_{n+1}}{a_n}\right | |x| \tag{1}$$

$$=\frac{1}{\lim\limits_{n\to\infty} \left |\frac{a_n}{a_{n+1}}\right |} |x|\tag{2}$$

$$=\frac{|x|}{R}\tag{3}$$

Then, for ##-R<x<R## we see that the expression in (3) is less than 1 and so the power series converges.
 
  • #3
I guess you've answered your own question.
 
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Likes chwala and zenterix

FAQ: Interval of convergence of power series from ratio test

What is the interval of convergence of a power series?

The interval of convergence of a power series is the range of values of the variable for which the series converges. It is determined by finding the values of the variable for which the series sums to a finite number.

How do you use the ratio test to determine the interval of convergence?

To use the ratio test, you take the limit of the absolute value of the ratio of successive terms in the power series. If the limit is less than 1, the series converges. This test can help identify the radius of convergence, which is then used to find the interval of convergence.

What is the radius of convergence and how is it related to the interval of convergence?

The radius of convergence is the distance from the center of the series to the boundary within which the series converges. Once the radius is found using the ratio test, the interval of convergence can be determined by testing the endpoints of the interval defined by the radius.

How do you test the endpoints of the interval of convergence?

To test the endpoints of the interval of convergence, you substitute the endpoint values into the original power series and check for convergence using other convergence tests such as the p-test, comparison test, or alternating series test. The series may converge at one endpoint, both endpoints, or neither.

What happens if the ratio test is inconclusive?

If the ratio test is inconclusive (i.e., the limit of the ratio of successive terms equals 1), other convergence tests need to be applied to determine the convergence at specific points. These tests can include the root test, comparison test, or integral test, among others.

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