Let f(x) = sum_{n=0}^{infty} x^n / n!. Let f^{(100)}(a) denote the 100th derivative of f evaluated at a. What is the val...

Question

Let f(x) = sum_{n=0}^{infty} x^n / n!. Let f^{(100)}(a) denote the 100th derivative of f evaluated at a. What is the value of f^{(100)}(0)? (Note: n! denotes factorial)
A. 1
B. C(100, 50)
C. 1/18
D. 1/100

Accepted Answer

1 The given power series sum_{n=0}^{infty} x^n/n! is exactly the Taylor series for e^x. Since e^x equals its own derivative, its 100th derivative is still e^x. Evaluating at x=0 gives e^0 = 1.

Let f(x) = sum_{n=0}^{infty} x^n / n!. Let f^{(100)}(a) denote the 100th derivative of f evaluated at a. What is the value of f^{(100)}(0)? (Note: n! denotes factorial) A. 1 B. C(100, 50) C. 1/18 D. 1/100