Investigate the convergence of ∫a^∞ (n^2 x e^-n^2 x^2)/(1+x^2) d x for a>0, and for a=0.

Name: Problem 7, Chapter 4: Measure, Integral and Probability
Uploaded: 2023-11-01T19:09:42-08:00
Duration: 3 min 30 s
Channel: William Semus
Description: Investigate the convergence of ∫a^∞ (n^2 x e^-n^2 x^2)/(1+x^2) d x for a>0, and for a=0.

Investigate the convergence of ∫a^∞ (n^2 x e^-n^2...

Key Concepts

Comparison Test for Integrals

The comparison test is a method used to determine the convergence or divergence of an integral by comparing it with another integral whose behavior is already known. By bounding the integrand with a function that decays faster (or has similar decay properties), one can infer the convergence properties of the original integral, especially when dealing with a product of an exponential decay and a rational function.

Behavior at Endpoints

Analyzing the behavior at the endpoints of the integration interval is essential for convergence analysis. At infinity, understanding how components like exponential decay control the integrand helps establish convergence. Near finite limits such as zero, approximating the integrand to see whether it has any singularity or behaves like a non-problematic function is necessary to determine if the integral converges from that side.

Improper Integrals

Improper integrals arise when one or both of the limits of integration are infinite or when the integrand has a singularity within the interval of integration. Studying their convergence involves replacing the problematic endpoints with finite limits and then taking a limit process. This is crucial when determining whether the integral yields a finite value or diverges.

Exponential Decay

Exponential decay refers to the rapid decrease of a function as its argument increases, typically seen in terms like e^(–?x²). This decay can dominate other factors in the integrand, ensuring that contributions from large values of the variable are negligible, which is a key factor in establishing the convergence of an integral over an infinite interval.

Transcript

00:01 So in this question, we're going to use the integral test to determine the convergence of the series, sum from n equals 1 to infinity, of n over n squared plus 1.

00:09 So what does the integral test say to do? it says i should convert my series into the associated improper integral.

00:18 So essentially, just replace the sigma with an integral, the integral from 1 to infinity of x over x squared plus 1 dx.

00:31 Now, since this is an improper integral, i start by rewriting it as a limit.

00:38 I have a limit as b approaches infinity of the integral from 1 to b of x over x squared plus 1 dx.

00:49 Now, in order to evaluate this, i am going to do a u substitution.

00:55 That's what i need to get my antiderivative.

00:59 So what am i going to let my u be? i'm going to let u equal x squared plus 1.

01:04 If that's my u, my du is equal to 2x dx.

01:10 And dividing by 2, i have du over 2 is equal to x dx.

01:18 Now, whenever i do a u substitution within a definite integral, i have to be sure to change my limits of integration.

01:27 So when i was in the world of x, my x limits of integration, they were from 1 to b.

01:34 How about u? well, if x is 1, my u would be 1 squared plus 1, which is 2.

01:41 And if x is b, my u would be b squared plus 1.

01:47 And so when i substitute in, what do i get? i get a limit as b approaches infinity of an integral from 2 to b squared plus 1.

02:01 My denominator, x squared plus 1, that becomes u.

02:05 In my numerator, i have x dx.

02:08 That becomes du over 2...

Please give Ace some feedback