Calculus II: Taylor & Maclaurin Series

Approximating Functions

A Taylor series allows us to represent complex functions (like sin(x) or e^x) as an infinite sum of polynomial terms. This is how calculators compute Sine and Cosine!

General Formula (centered at a):

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

Maclaurin Series (centered at 0):

A special case where a = 0. Example for e^x:

e^x ≈ 1 + x + x^2/2! + x^3/3! + ...

Ratio Test Application

Not all series converge for all x. To find the Interval of Convergence, we typically use the Ratio Test.

The Logic:

1. Take the limit as n approaches infinity of |a_(n+1) / a_n|.
2. Set the result < 1 to find the values of x where it converges.
3. Check Endpoints: Always plug the endpoints back into the original series to check for conditional convergence.

Homework Help: Use our AI Math Solver to check your ratio test steps if you keep getting 'Infinity'.