Calculus II: Taylor & Maclaurin Series

Approximating Functions

                <p>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!</p>

General Formula (centered at a):

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

Maclaurin Series (centered at 0):

                <p>A special case where a = 0. Example for e^x:</p>
                <pre>e^x ≈ 1 + x + x^2/2! + x^3/3! + ...</pre>

Ratio Test Application

                <p>Not all series converge for all x. To find the <b>Interval of Convergence</b>, we typically use the Ratio Test.</p>