of the Taylor series of ,f(x), centered
at ,c, by finding the coefficients of the first few
powers of x and looking for a pattern. (The formulas for
several of these are found in Key Idea 9.7.11; show work
verifying these formula.)
- In The Following Exercise Find A Formula For The Nth Term Of The Taylor Series Of F X Centered At C By Finding The 1 (4.09 KiB) Viewed 25 times
- In The Following Exercise Find A Formula For The Nth Term Of The Taylor Series Of F X Centered At C By Finding The 2 (193.96 KiB) Viewed 25 times
Key Idea 9.7.11. Important Taylor Series Expansions. Function and Series First Few Terms Interval of Convergence 1+2+ x2 23 + 2! + 3! (-00,00) IM8 iM8 = x2n+1 23 - + x 5 5! x 7 + 7! (-00,00) 3! 72 2 26 sin(x) = (-1)" (2n + 1)! x2n cos(x) = (-1) (2n)! In(x) = (-1)n+1 (x - 1)" 1 + 24 4! +... (-00,00) 2! n=0 6! = (x - 1) (x - 1)2 + 2 (x - 1)3 3 (0, 2) n n=1 oo 1 C' in 1+ x + x2 + x3 +.. 1-2 (-1,1) n=0 x2n+1 X3 tan-+(x) = (-1)" 25 + 5 27 7 +... [-1,1] 2n +1 3 n=0 (1+2)* = (%)" + x)=Σ n 2 k(k – 1) 1+ kx + 2! 2 +... (-1,1) n=0 1 Note that for (1 + x)\, the interval of convergence may contain one or both endpoints, depending on the value of k, and we are using the generalized binomial coefficients () k(k – 1)... (k – (n − 1)) = n n! in-context