Unformatted text preview: information about the function. So far, we have seen only those examples that result from manipulation of our one fundamental example, the geometric series. We would like to start with a given function and produce a series to represent it, if possible. Suppose that f ( x ) = ∞ X n =0 a n x n on some interval of convergence. Then we know that we can compute derivatives of f by taking derivatives of the terms of the series. Let’s look at the ﬁrst few in general: f ± ( x ) = ∞ X n =1 na n x n1 = a 1 + 2 a 2 x + 3 a 3 x 2 + 4 a 4 x 3 + ··· f ±± ( x ) = ∞ X n =2 n ( n1) a n x n2 = 2 a 2 + 3 · 2 a 3 x + 4 · 3 a 4 x 2 + ··· f ±±± ( x ) = ∞ X n =3 n ( n1)( n2) a n x n3 = 3 · 2 a 3 + 4 · 3 · 2 a 4 x + ···...
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 Spring '07
 JonathanRogawski
 Math, Calculus, Power Series, Taylor Series, Mathematical Series, power series representation

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