Taylor series of f(x)=e^x centered at 0

Question

Find the 1st, 2nd, and 3rd degree Taylor polynomials of f(x)=e^x centred at 0. If you can guide me through the thought process of getting the answers.

My current guesses are:

T₁(x)= 1+x

T₂(x)= 1+x + x^2/2!

T₃(x)= 1+ x + x^2/2! + x^3/3!

Answer 1

Taylor series for f(x) = e^x, centered at 0

1+x

1+x +x^2/2!

1+x +x^3/3!

Taylor series centered at 0 is the MacLaurin series

=f(a)(x-a)^0/0! + f'(a)(x-a)^1/1! + f"(a)(x-a)^2/(x-a)! + f"'(a)(x-a)^3/(x-a)! + ...

= f(0)x^0/0! +(e^0)x + (e^0)x^2/2 + (e^0)x^3/6 + ....

(all derivatives of e^x = e^x. e^0=1, all coefficients of the series =1)

=1 +x + x^2/2 + x^3/6 + ...

e=about 2.718281828

1+1+1/2 +1/6 = about 2.666666

error of about 0.05161518