EXAMPLE 1Find the Maclaurin series of the function f(x) = e8x and its radius of convergence.If f(x) = e3x, then f(m)(x) = 8"e8x, so f(n(0) = 8n@0 .SOLUTIONfor all n. Thereforethe Taylor series for f at 0 (that is, the Maclaurin series) iscoΣn = 0n!n!n = 064x512x38x= 1 ++1!+...2!3!To find the radius of convergence we let a,= 8"x"/n!. ThencalcFOperat8n+1+1a,n!n + 1Functican8"xn(n + 1)!Symba8|x|Relatio< 1n + 1Setsso, by the Ratio Test, the series converges for all x and the radius of convergence is R = o.Vi 0!VectorsTrigGreek

Explanation of the answer is as follows

EXAMPLE 1 Find the Maclaurin series of the function f(x) = e8x and its radius of convergence. SOLUTION If f(x) = e8x, then f((x) 8"e&r so f(n)(0) = 8"e0 for all n. Therefore = the Taylor series for f at 0 (that is, the Maclaurin series) is n! n! n = 0 n = 0 8x = 1 + 1! 64r2 2! 512x3 +... 3! To find the radius of convergence we let a, = 8"x"/n!. Then calch Operat 8" +1,n+1 an+ 1 n! Functio an 8"x" X (n + 1)! Symbo 8|x|| n + 1 Relatic < 1 Sets so, by the Ratio Test, the series converges for all x and the radius of convergence is R = ∞. o! Vector: Trig

EXAMPLE 1 Find the Maclaurin series of the function f(x) = e8x and its radius of convergence. SOLUTION If f(x) = e8x, then f((x) 8"e&r so f(n)(0) = 8"e0 for all n. Therefore = the Taylor series for f at 0 (that is, the Maclaurin series) is n! n! n = 0 n = 0 8x = 1 + 1! 64r2 2! 512x3 +... 3! To find the radius of convergence we let a, = 8"x"/n!. Then calch Operat 8" +1,n+1 an+ 1 n! Functio an 8"x" X (n + 1)! Symbo 8|x|| n + 1 Relatic < 1 Sets so, by the Ratio Test, the series converges for all x and the radius of convergence is R = ∞. o! Vector: Trig

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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Question

EXAMPLE 1
Find the Maclaurin series of the function f(x) = e8x and its radius of convergence.
If f(x) = e3x, then f(m)(x) = 8"e8x
, so f(n(0) = 8n@0 .
SOLUTION
for all n. Therefore
the Taylor series for f at 0 (that is, the Maclaurin series) is
co
Σ
n = 0
n!
n!
n = 0
64x
512x3
8x
= 1 +
+
1!
+...
2!
3!
To find the radius of convergence we let a,
= 8"x"/n!. Then
calcF
Operat
8n+1+1
a,
n!
n + 1
Functic
an
8"xn
(n + 1)!
Symba
8|x|
Relatio
< 1
n + 1
Sets
so, by the Ratio Test, the series converges for all x and the radius of convergence is R = o.
Vi 0!
Vectors
Trig
Greek

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