Infinite Series Function problem?

Question

p(x)= 1 + x + (1/2!)x^2 + (1/3!)x^3 + (1/4!)x^4 ...

Differentiate the series (the first 7 terms), what do you notice?

What function must the series be modeling?

Answer 1

this is e^x

Answer 2

The series is modelling exp(x) = e^x

= 2.7182818^x and the derivative of

that function is the same exp(x), so

without calculating i know that we obtain

the same series after differentiation.

Answer 3

p(x)= 1 + x + (1/2!)x^2 + (1/3!)x^3 + (1/4!)x^4

dp/dx = 0 + 1 + 2/2! x + 3/3! x² + 4/4! x³ + ...

Notice that n/n! = 1/(n - 1)! for n >= 1

so:

dp/dx = 1 + x + 1/2x² + 1/3x³ + ...

so you get the same result as before.

This is a power series for the exponential function, exp(x)