Cal II taylor maclaurin series, GBT, Bounded error

What is a Taylor Series?

A Taylor series is a representation of a function as an infinite sum of terms, calculated from the values of the function's derivatives at a single point (the 'center' of the series). For a function f(x) centered at a, the Taylor series is given by:f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!} (x-a)^n = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \dots

What is a Maclaurin Series?

A Maclaurin series is a special case of a Taylor series where the series is centered at a=0. It is given by:f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!} x^n = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \dots

State the Generalized Binomial Theorem.

The Generalized Binomial Theorem extends the binomial theorem to cases where the exponent is any real number (k), not just a non-negative integer. For |x|<1, it states: (1+x)^k = \sum_{n=0}^{\infty} \binom{k}{n} x^n = 1 + kx + \frac{k(k-1)}{2!}x^2 + \frac{k(k-1)(k-2)}{3!}x^3 + \dotswhere the generalized binomial coefficient is defined as: \binom{k}{n} = \frac{k(k-1)\dots(k-n+1)}{n!}. The radius of convergence for this series (when k is not a non-negative integer) is R=1.

How can you bound the error of a Taylor Polynomial approximation?

The error when approximating a function f(x) with its n-th degree Taylor polynomial Pn(x) centered at a is given by the Taylor Remainder Rn(x) = f(x) - Pn(x). According to Taylor's Theorem (Lagrange form of the remainder), there exists a number c between a and x such that:Rn(x) = \frac{f^{(n+1)}(c)}{(n+1)!} (x-a)^{n+1}To bound the error, one must find an upper bound M for |f^{(n+1)}(c)| on the interval between a and x. Then, the error bound is:|R_n(x)| \le \frac{M}{(n+1)!} |x-a|^{n+1}

What is the radius of convergence for a power series?

The radius of convergence, R, is a non-negative number or \infty such that the power series converges for |x-a| < R and diverges for |x-a| > R. At the endpoints |x-a| = R, the series may converge or diverge.

List two common Maclaurin series expansions.

1. Exponential function: e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!} = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \dots, for all x (R=\infty).

2. Sine function: \sin(x) = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!} = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \dots, for all x (R=\infty).

How do you find the Maclaurin series for a function (f(x)) given its derivatives?

To find the Maclaurin series for f(x), evaluate the function and its successive derivatives at x=0. Then, substitute these values into the Maclaurin series formula:f(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \frac{f'''(0)}{3!}x^3 + \dots

What is the Alternating Series Estimation Theorem, and when can it be used to bound error?

For an alternating series \sum (-1)^n bn where bn > 0, {bn} is a decreasing sequence, and bn \to 0, the Alternating Series Estimation Theorem states that the absolute value of the error Rn (the remainder after summing the first n terms) is less than or equal to the absolute value of the first neglected term: \left|Rn\right| \le b_{n+1}. It can only be used for alternating series that satisfy these conditions.

If k is a non-negative integer, what does the Generalized Binomial Theorem reduce to?

If k is a non-negative integer, the Generalized Binomial Theorem reduces to the classical Binomial Theorem: (1+x)^k = \sum_{n=0}^{k} \binom{k}{n} x^n. The series becomes a finite sum because the binomial coefficient \binom{k}{n} becomes zero when n > k. (\binom{k}{n} = \frac{k(k-1)\dots(k-n+1)}{n!} so if n > k, one of the terms in the numerator (k-k=0) will be zero).

Explain how to use known Maclaurin series to derive new ones through substitution or differentiation.

1. Substitution: Replace x in a known series with a function of x (e.g., to find the series for e^{-x^2}, substitute -x^2 into the series for e^x).

2. Differentiation: Differentiate a known power series term-by-term to find the series for its derivative (e.g., differentiate the series for \sin(x) to get the series for \cos(x)).

When does a Taylor series converge to the original function f(x)?

A Taylor series for f(x) centered at a converges to f(x) if and only if the remainder term Rn(x) = f(x) - Pn(x) approaches zero as n \to \infty for all x in the interval of convergence. Functions for which this is true are called analytic functions.

What is the significance of the point a in a Taylor series expansion?

The point a is the 'center' of the Taylor series expansion. It is the point around which the function's behavior is being approximated. The accuracy of the Taylor polynomial approximation is generally best near x=a and may decrease as x moves further away from a.

Give an example of a function whose Maclaurin series converges for all x.

The Maclaurin series for e^x, \sin(x), and \cos(x) all converge for all real values of x. Their radius of convergence is R=\infty.

How can the Generalized Binomial Theorem be used to approximate values?

The Generalized Binomial Theorem can be used to approximate values by taking the first few terms of the series and evaluating them at a specific x value, typically for \left|x\right|<1. For example, to approximate \sqrt{1.01}, one could use (1+x)^k where x=0.01 and k=1/2 and sum the first few terms of its binomial series.

What is the radius of convergence for a power series?

The radius of convergence, R, is a non-negative number or \infty such that the power series converges for \left|x-a\right| < R and diverges for \left|x-a\right| > R. At the endpoints \left|x-a\right| = R, the series may converge or diverge.\n\n

What is the interval of convergence for a power series?

The interval of convergence is the set of all x values for which a power series converges. It is centered at a. It is typically of the form (a-R, a+R), [a-R, a+R], (a-R, a+R], or [a-R, a+R). For R=\infty, the interval is (-\infty, \infty), and for R=0, it's just the point x=a. The endpoints must be checked separately for convergence.\n\n

After finding the radius of convergence R, how do you determine the interval of convergence?

1. Identify the open interval: The series converges on the open interval (a-R, a+R).\n2. Check the endpoints: Substitute x = a-R and x = a+R back into the original power series. Test each endpoint using a suitable convergence test (e.g., p-series test, alternating series test, comparison test, divergence test). Include the endpoint in the interval of convergence if the series converges at that point.\n3. Form the final interval: Combine the open interval with any converging endpoints to form the final interval of convergence (e.g., (a-R, a+R], [a-R, a+R]).