(-1)"x2n+ Given the power series > n=0 (2n + 1)n! |an+1 (a) Simplify an and find the limit. State the radius of convergence and the end- points. If applicable, substitute to show whether each endpoint is in the interval of convergence or not.
(-1)"x2n+ Given the power series > n=0 (2n + 1)n! |an+1 (a) Simplify an and find the limit. State the radius of convergence and the end- points. If applicable, substitute to show whether each endpoint is in the interval of convergence or not.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![2. Given the power series
\[
\sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!}
\]
(a) Simplify \(\left| \frac{a_{n+1}}{a_n} \right|\) and find the limit. State the radius of convergence and the endpoints. If applicable, substitute to show whether each endpoint is in the interval of convergence or not.
(b) It can be shown that the series converges to \(f(x) = \int_0^x e^{-t^2} \, dt\) on its interval of convergence. To illustrate this, find \(s_{10}\), \(s_{20}\), and \(s_{30}\). Plot these three polynomials and \(f\) on the same set of axes in the window \(x \in [-10, 10]\), \(y \in [-2, 2]\).
(c) Notice that \(\int_0^{\infty} e^{-t^2} \, dt\) cannot be integrated using standard techniques, but the series can be used to approximate values of the definite integral in \(f\) for any value of \(x\). Use \(s_{100}\) to obtain a decimal approximation for \(f(5)\) (NOTE from the graph that this is a pretty good approximation for \(\int_0^{\infty} e^{-t^2} \, dt\)). Compare your answer with the decimal approximation for \(\frac{\sqrt{\pi}}{2}\). What do you notice?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc6d45b7c-770b-4dbe-b3cf-74fe1dfded8d%2F398fc581-de21-4b95-8cf1-fb6dcb62e90f%2F20tuewa_processed.png&w=3840&q=75)
Transcribed Image Text:2. Given the power series
\[
\sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!}
\]
(a) Simplify \(\left| \frac{a_{n+1}}{a_n} \right|\) and find the limit. State the radius of convergence and the endpoints. If applicable, substitute to show whether each endpoint is in the interval of convergence or not.
(b) It can be shown that the series converges to \(f(x) = \int_0^x e^{-t^2} \, dt\) on its interval of convergence. To illustrate this, find \(s_{10}\), \(s_{20}\), and \(s_{30}\). Plot these three polynomials and \(f\) on the same set of axes in the window \(x \in [-10, 10]\), \(y \in [-2, 2]\).
(c) Notice that \(\int_0^{\infty} e^{-t^2} \, dt\) cannot be integrated using standard techniques, but the series can be used to approximate values of the definite integral in \(f\) for any value of \(x\). Use \(s_{100}\) to obtain a decimal approximation for \(f(5)\) (NOTE from the graph that this is a pretty good approximation for \(\int_0^{\infty} e^{-t^2} \, dt\)). Compare your answer with the decimal approximation for \(\frac{\sqrt{\pi}}{2}\). What do you notice?
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