3 n n0 11 as well as the trigonometric functions cos

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Unformatted text preview: s include the exponential function ∞ ex = 1 + x + 1n 12 1 x + x3 + · · · = x, 2! 3! n! n=0 (1.1) as well as the trigonometric functions cos x = 1 − 12 1 x + x4 − · · · = 2! 4! and sin x = x − 13 1 x + x5 − · · · = 3! 5! ∞ (−1)k k=0 ∞ (−1)k k=0 1 2k x (2k )! 1 x2k+1 . (2k + 1)! An infinite series of this type is called a power series. To be precise, a power series centered at x0 is an infinite sum of the form ∞ a0 + a1 (x − x0 ) + a2 (x − x0 )2 + · · · = an (x − x0 )n , n=0 where the an ’s are constants. In advanced treatments of calculus, these power series representations are often used to define the exponential and trigonometric functions. Power series can also be used to construct tables of values for these functions. For example, using a calculator or PC with suitable software installed (such as Mathematica), we could calculate 2 1+1+ 12 1n 1= 1 = 2.5, 2! n! n=0 1 4 1+1+ 1n 12 1 1 1 + 13 + 14 = 1 = 2.70833, 2! 3! 4! n! n=0 8 12 1n 1 = 2.71806, n! n=0 1n 1 = 2.71828...
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