Question

Find the Maclaurin series for $f(x)$ using the definition of a Maclaurin series. [Assume that $f$ has a power series expansion. Do not show that $R_n(x) \to 0$.] $f(x) = e^{-5x}$ $f(x) = \sum_{n=0}^{\infty} (\text{_})$ Find the associated radius of convergence R. $R = \text{_}$

Name: find the maclaurin series for fx using the definition of a maclaurin series assume that f has a power series expansion do not show that rnx to 0 fx e 5x fx sumn0infty text find the as 67034
Uploaded: 2021-09-07T21:37:21-08:00
Duration: 1 min 45 s
Channel: Adrian Co
Description: find the maclaurin series for fx using the definition of a maclaurin series assume that f has a power series expansion do not show that rnx to 0 fx e 5x fx sumn0infty text find the as 67034

          Find the Maclaurin series for $f(x)$ using the definition of a Maclaurin series. [Assume that $f$ has a power series expansion. Do not show that $R_n(x) \to 0$.]
$f(x) = e^{-5x}$
$f(x) = \sum_{n=0}^{\infty} (\text{_____})$
Find the associated radius of convergence R.
$R = \text{_____}$

find the maclaurin series for fx using the definition of a maclaurin series assume that f has a power series expansion do not show that rnx to 0 fx e 5x fx sumn0infty text find the as 67034

Added by Joseph M.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Adrian Co

09/07/2021

Step 1

Then $f(0) = e^0 = 1$ $f'(x) = -5e^{-5x}$, $f'(0) = -5$ $f''(x) = 25e^{-5x}$, $f''(0) = 25$ $f'''(x) = -125e^{-5x}$, $f'''(0) = -125$ In general, $f^{(n)}(x) = (-5)^n e^{-5x}$, so $f^{(n)}(0) = (-5)^n$. Therefore, the Maclaurin series is $$ f(x) = Show more…

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Find the Maclaurin series for $f(x)$ using the definition of a Maclaurin series. [Assume that $f$ has a power series expansion. Do not show that $R_n(x) \to 0$.] $f(x) = e^{-5x}$ $f(x) = \sum_{n=0}^{\infty} (\text{_____})$ Find the associated radius of convergence R. $R = \text{_____}$