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Unformatted text preview: 8.9  More stuﬀ you can do with power series
You should have the following Maclaurin series memorized. ∞ e x =
n=0 ∞ xn n! (−1)n x2n+1 (2n + 1)! (−1)n x2n (2n)! xn sin x =
n=0 ∞ cos x = 1 = 1−x
n=0 ∞ n=0 We can ﬁnd new Maclaurin series via substitution into the above formulas. Examples Find the Maclaurin series’ for the following functions. 1. f (x) = xex
3 2. f (x) = 1 1 + 3x 1 Diﬀerentiation and Integration via power series
We can also use diﬀerentiation and integration of power series to help ﬁnd power series for functions. Theorem. TermByTerm Diﬀerentiation and Integration of Power Series
∞ If a power series
n=0 cn (x − a)n converges for a − R < x < a + R, R > 0, it deﬁnes a function on its interval of convergence
∞ f ( x) =
n=0 cn (x − a)n . f (x) has derivatives and antiderivatives as follows. 2 Examples Find the Maclaurin series’ of the following functions.
x 1. F (x) =
0 cos(t3 )dt 2. f (x) = 1 (1 + x)2 3 Finding Limits with Power Series Power series also oﬀer an alternative way to calculate limits of functions as x → a, where a the center of the series. 1 − cos t − t2 /2 t→0 t4 lim 4 ...
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 Spring '03
 MECothren
 Maclaurin Series, Multivariable Calculus, Power Series

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