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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. Term-By-Term 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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