The
Taylor Series and Maclaurin Series (Continued)
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Since the series
expansions are power series, you can determine the interval of convergence for each
expansion by using the ratio test (as before). Click here for a reminder
of the ratio test applied to power series. i.e. Interval of convergence for expansion of ex is (
-∞, ∞ ) Interval of convergence for expansion
of sin x is (
-∞, ∞ ) |
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Example: Evaluate the following limit
using an appropriate series expansion.
lim (sin x2 - x2
) / x6 (Check
using l’Hopital’s
rule) x → 0 From 7 above (replace x
with x2 ) sin x2 = x2 - x6/3! + x10/5!
lim (sin x2 - x2
) / x6 = -
3! + x4/5! - . . = -1/6
(result) x → 0 |
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Example: Evaluate the following integral (use 4
terms in the expansion for e√x ) 1 I
= ∫ e√x
dx 0 Replace
x with √x ) in the expansion
for ex giving
e√x =
1 + √x
+ x/2! + x3/2/3! 1 I
= ∫ [1 + √x +
x/2! + x3/2/3! ] dx =
119/60 (result) 0 |
Copyright © 2017 Richard C. Coddington
All rights reserved.