17. + 1 points SCalcET6 11.10·AE.02. My Notes Ask Video Example EXAMPLE 2 Prove
ID: 2891006 • Letter: 1
Question
17. + 1 points SCalcET6 11.10·AE.02. My Notes Ask Video Example EXAMPLE 2 Prove that 7e is equal to the sum of its Maclaurin series Tutorial Online Textbook SOLUTION If rx) = 7e", then f (n+1)(x)- for all n. If d is any positive number and | x I s d, then I f*)(x) | - 7ed. So Taylor's Inequality, with a = 0 and M-7ed says that I x for | x I sd Notice that the same constant M = 7ed works for every value of n. But we have 7ed lim It follows from the Squeeze Theorem that limn“ | Rn(x)-0 and therefore limn Rn(x)- for all values of x. By Theorem 8, 7e is equal to the sum of its Maclaurin series, that is for allx n! n=0Explanation / Answer
Solution:
Given Series
= 7ex
The series of the function f(x) at a is defined as
f(x) = f(a) + f'(a)/1! (x-a) + f"(a)/2! (x-a)2 + f"'(a)/3!(x-a)3 + ......
Maclaurin Series of function f(x) is a Taylor series of function f(x) at : a = 0
f(x) = f(0) + f'(0)/1! (x) + f"(0)/2! (x)2 + f"'(0)/3!(x)3 + ......
Find the derivatives of f(x) = 7ex , at a = 0
f(x) = 7e0 + f'(7ex)(0)/1! (x) + f"(7ex)(0)/2! (x)2 + f"'(7ex)(0)/3!(x)3 + ......
7e0 = 7
d/dx(7ex)(0) = 7ex
at x = 0
= 7 e0
= 7 * 1
= 7
d2/dx2(7ex)(0) = 7
d3/dx3(7ex)(0) = 7
d4/dx4(7ex)(0) = 7
= 7 + 7/1! x + 7/2! x2 + 7/3! x3 + 7/4! x4 + .......
= 7 + 7x + 7/2 x2 + 7/6 x3 + 7/24 x4 + .......
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