Calculus II — Taylor & Maclaurin Series — 3-topic bundle

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>

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You know that polynomials are easy to work with (differentiate, integrate, evaluate). The goal of a Taylor series is to represent a more complex function, say $f(x)$, as an infinite polynomial:

$f(x) = c_0 + c_1(x-a) + c_2(x-a)^2 + c_3(x-a)^3 + \dots$

The key is to find the coefficients $c_0, c_1, c_2, \dots$ so that the polynomial matches $f(x)$ and all its derivatives at a specific "center" point, $x=a$.

When you approximate a function $f(x)$ with its $n$-th degree Taylor polynomial $P_n(x)$ centered at $a$, the difference between them is called the remainder or error, $R_n(x) = f(x) - P_n(x)$. Error bounds help us find an upper limit for the absolute value of this remainder, i.e., $|R_n(x)| \le \text{bound}$.