I'm having trouble with series.

Question

Let f(x)= ∑^∞_n=0 (x^n)/(n!)

Find f(0)= 1

f(1)= e

f'(x)= f(x)

f''(x)= f(x)

Not exactly sure how we're supposed to go about finding those answers, or how to start.

Answer 1

The series ∑_{(n=0 to ∞)} [xⁿ/n!] is the Maclaurin series for e^x and has interval of convergence ∞.

 

So, f(x) = e^x.

 

f(0) = e⁰ = 1

f(1) = e¹ = e

f'(x) = e^x = f(x)

f"(x) = e^x = f(x)