For what function?

Question

∑ ((-1)^n*x^n)/n! is the Taylor series about zero for what function?

 

Answer: e^-x

 

 

Answer 1

∑ (x^n)/n! = e^x, n from 0 to ∞

 

∑ ((-1)^n*x^n)/n!, n from 0 to ∞

= ∑ ((-x)^n)/n!, n from 0 to ∞

= e^-x

Answer 2

Taylor Series at x = 0 is the Maclaurin Series.

 

It says that a function f(x) can be approximated by an infinite series:

f(x) = f(0) + f'(0) * x + f''(0) * x^2/2! + f'''(0) * x^3/3! + ...

 

If f(x) = e^x then all the derivatives are also e^x and at x = 0 are all 1.

 

So f(x) = 1 + x + x^2/2! + x^3/3! + ... = sum[n=0 to inf][x^n/n!] = e^x.

 

f(-x) = 1 - x + x^2/2! - x^3/3! + ... = sum[n=0 to inf][(-1)^n * x^n/n!] = e^(-x).