ISPA - Infinite Series and Polynomial Approximations Module Overview

Their Infinite Series and Polynomial Approximations Module Overview

Introduction

The importance of infinite series in calculus stems from Isaac Newton's (one of the cofounders of modern calculus) idea of representing functions as sums of infinite series. Many tests have been developed to determine if an infinite series has a sum and thus converges to a limiting value. Power series are of particular importance in calculus as they provide an efficient means of evaluating integrals without explicit antiderivatives. Lest you worry about the accuracy of approximating a function with a Taylor polynomial, the Lagrange error bound can measure the error associated with the approximation.

Essential Questions

How do you find the sum of an infinite geometric series?
How are properties of geometric series used in applications?
What does it mean for a series to converge or diverge?
How does the harmonic series relate to a p-series?
What are the Maclaurin series expressions for the functions e^x, sin x, cos x, and 1/(1-x)?
How can technological tools be used to explore convergence and divergence of series?
What is a Taylor polynomial approximation?
What is the difference between a Taylor series and Maclaurin series?
What role do remainders/error bounds play when approximating series sums and polynomials?
How do you know which test for convergence or divergence to use?
How do the terms of a series relate to its area under a curve and to its corresponding improper integral?

Key Terms

The following key terms will help you understand the content in this module.

Absolute convergence - A series $LaTeX: \sum a_n$ $\sum a_n$ is absolutely convergent if the series of absolute values $LaTeX: \sum\left|a_n\right|$ $\sum\left|a_n\right|$ is convergent.

Alternating series - An infinite series in which the terms are alternately positive and negative.

Alternating series estimation theorem - If $LaTeX: \sum\left(-1\right)^{n+1}a_n$ $\sum\left(-1\right)^{n+1}a_n$ is the sum of an alternating series that satisfies 0 ≤ a_n+1 ≤ a_n and $LaTeX: \lim_{n\to\infty}a_n=0$ $\lim_{n\to\infty}a_n=0$ , then the absolute value of the remainder Rn involved in approximating the sum s by s_n is less than or equal to the numerical value of the first unused term, i.e., |R_n|=|s-s_n| ≤ a_n+1.

Alternating series test - For a_n > 0, if the series $LaTeX: s=\sum_{n=1}^{\infty}\left(-1\right)^{n+1}a_n$ $s=\sum_{n=1}^{\infty}\left(-1\right)^{n+1}a_n$ satisfies

i) a_n+1 ≤ a_n for all n

ii) $LaTeX: \lim_{n\to\infty}a_n=0$ $\lim_{n\to\infty}a_n=0$

Then the series converges.

Bounded sequence - A sequence {a_n} that is bounded above and below. A sequence {a_n} is bounded above if there is a number M such that a_n ≤ M for all n ≥ 1. It is bounded below if there is a number m such that m ≤ a_n for all n ≥ 1.

Comparison test - Suppose that $LaTeX: \Sigma a_n$ $\Sigma a_n$ and $LaTeX: \Sigma b_n$ $\Sigma b_n$ are series with positive terms.

i) If $LaTeX: \Sigma b_n$ $\Sigma b_n$ is convergent and a_n ≤ b_nfor all n, then $LaTeX: \Sigma a_n$ $\Sigma a_n$ is also convergent

ii) If $LaTeX: \Sigma b_n$ $\Sigma b_n$ is divergent and a_n ≥ b_n for all n, then $LaTeX: \Sigma a_n$ $\Sigma a_n$ is also divergent.

Conditional convergence - A series $LaTeX: \Sigma a_n$ $\Sigma a_n$ is conditionally convergent if it is convergent but not absolutely convergent.

Convergence of a series - When the sequence of partial sums converges to a real number.

Converges - Approaching a finite limiting value.

Divergence of a series - When the sequence of partial sums diverges.

Diverges -Having no finite limit.

Geometric series - A series of the form S = a + ar + ar² + ... + ar^n-1 + ... +... = $LaTeX: \sum_{n=1}^{\infty}ar^{n-1}$ $\sum_{n=1}^{\infty}ar^{n-1}$ , where r is the nonzero constant ratio between any two consecutive terms and a ≠ 0.

Harmonic series - An infinite series in which the reciprocals of the terms form the arithmetic sequence 1 + 1/2 + 1/3 + 1/4 + ... = $LaTeX: \sum_{n=1}^{\infty}\frac{1}{n}$ $\sum_{n=1}^{\infty}\frac{1}{n}$ .

Integral test - Suppose f is a continuous, positive, decreasing function on [1, ∞) and let a_n = f(n). Then the series $LaTeX: \sum_{n=1}^{\infty}$ $\sum_{n=1}^{\infty}$

and the integral $LaTeX: \int_1^{\infty}f\left(x\right)dx$ $\int_1^{\infty}f\left(x\right)dx$

both converge or both diverge.

Interval of convergence - The set of all values of x for which the power series $LaTeX: \sum_{n=0}^{\infty}\left(x-a\right)^n$ $\sum_{n=0}^{\infty}\left(x-a\right)^n$ centered at a converges.

Lagrange error bound - The error associated with approximating a function value f(x) by a Taylor polynomial P_n satisfies the inequality $LaTeX: \left|R_n\left(x\right)\right|\frac{\left(x-c\right)}{\left(n+1\right)!}\max\left|f^{\left(n+1\right)}\left(z\right)\right|$ $\left|R_n\left(x\right)\right|\frac{\left(x-c\right)}{\left(n+1\right)!}\max\left|f^{\left(n+1\right)}\left(z\right)\right|$ , where max $LaTeX: \left|f^{\left(n+1\right)}\left(z\right)\right|$ $\left|f^{\left(n+1\right)}\left(z\right)\right|$ is the maximum value of $LaTeX: f^{\left(n+1\right)}\left(z\right)$ $f^{\left(n+1\right)}\left(z\right)$ between x and c.

Lagrange form of the remainder - The remainder in Taylor\'s formula, $LaTeX: R_n\left(x\right)\le\frac{f^{\left(n+1\right)}\left(c\right)}{\left(n+1\right)!}\left(x-a\right)^{n+1}$ $R_n\left(x\right)\le\frac{f^{\left(n+1\right)}\left(c\right)}{\left(n+1\right)!}\left(x-a\right)^{n+1}$ .

Limit comparison test - Suppose $LaTeX: \Sigma a_n$ $\Sigma a_n$ and $LaTeX: \Sigma b_n$ $\Sigma b_n$ are series with positive terms. If $LaTeX: \lim_{n\to\infty}\frac{a_n}{b_n}=c$ $\lim_{n\to\infty}\frac{a_n}{b_n}=c$ , where c is a finite number and c > 0, then either both series converge or both diverge.

Limit of a sequence - The limiting value of {a_n} and defined as a real number L, written as $LaTeX: \lim_{n\to\infty}a_n=L$ $\lim_{n\to\infty}a_n=L$

or $LaTeX: a_n\longrightarrow L\:as\:n\longrightarrow\infty$ $a_n\longrightarrow L\:as\:n\longrightarrow\infty$ , if the terms a_ncan be made sufficiently close to L by taking n sufficiently large.

Maclaurin series - A power series for any function f with infinitely many derivatives throughout some interval containing 0 as an interior point and of the form $LaTeX: \sum_{n=0}^{\infty}\frac{f^{\left(n\right)}\left(0\right)}{n!}\left(x^n\right)$ $\sum_{n=0}^{\infty}\frac{f^{\left(n\right)}\left(0\right)}{n!}\left(x^n\right)$ .

Monotonic sequence - A sequence that is either increasing or decreasing.

Monotonic sequence theorem - Every bounded, monotonic sequence is convergent.

nth-term test for divergence - If $LaTeX: \lim_{n\to\infty}a_n$ $\lim_{n\to\infty}a_n$ does not exist or if $LaTeX: \lim_{n\to\infty}a_n\ne0$ $\lim_{n\to\infty}a_n\ne0$ , then $LaTeX: \sum_{n=0}^{\infty}a_n$ $\sum_{n=0}^{\infty}a_n$ diverges.

p-series - A generalization of the harmonic series where $LaTeX: \sum_{n=1}^{\infty}\frac{1}{n^P}$ $\sum_{n=1}^{\infty}\frac{1}{n^P}$

p-series test - The p-series $LaTeX: \sum_{n=1}^{\infty}\frac{1}{n^P}$ $\sum_{n=1}^{\infty}\frac{1}{n^P}$ is convergent if p > 1 and divergent if p ≤ 1.

Power series - An infinite series of the form

$LaTeX: \sum_{n=0}^{\infty}c_nx^n=c_0+c_1x+c_2x^2+c_3x^3+...$ $\sum_{n=0}^{\infty}c_nx^n=c_0+c_1x+c_2x^2+c_3x^3+...$

where x is a variable and c_n's are constants (coefficients).

Radius of convergence - The radius of the interval around a given point a such that a given power series $LaTeX: \sum_{n=0}^{\infty}c_n\left(x-a\right)^n$ $\sum_{n=0}^{\infty}c_n\left(x-a\right)^n$ converges absolutely at all points inside the interval.

Ratio test - Let $LaTeX: \sum a_n$ $\sum a_n$ be a series with nonzero terms.

i) If $LaTeX: \lim_{n\to\infty}\left|\frac{a_n+1}{a_n}\right|<1$ $\lim_{n\to\infty}\left|\frac{a_n+1}{a_n}\right|<1$ , then $LaTeX: \sum a_n$ $\sum a_n$ converges absolutely

ii)If $LaTeX: \lim_{n\to\infty}\left|\frac{a_n+1}{a_n}\right|>1\:or\:\lim_{n\to\infty}\left|\frac{a_n+1}{a_n}\right|=\infty$ $\lim_{n\to\infty}\left|\frac{a_n+1}{a_n}\right|>1\:or\:\lim_{n\to\infty}\left|\frac{a_n+1}{a_n}\right|=\infty$ , then $LaTeX: \sum a_n$ $\sum a_n$ diverges.

iii) If $LaTeX: \lim_{n\to\infty}\left|\frac{a_n+1}{a_n}\right|=1$ $\lim_{n\to\infty}\left|\frac{a_n+1}{a_n}\right|=1$ , then the Ratio test is inconclusive.

Sequence - An infinite list of numbers written in a definite order and of the general form a₁, a₂, a₃, a₄ ..., a_n,, a_n+1 ... . The notation {a_n} refers to a sequence whose n^th term is a_n.

Taylor polynomial - Let f be a function whose first n derivatives exist at x = c. Then the polynomial $LaTeX: P_n\left(x\right)=f\left(c\right)+f'\left(c\right)\left(x-c\right)+\frac{f^n\left(c\right)}{2!}\left(x-c\right)^2+...+\frac{f^n\left(c\right)}{n!}\left(x-c\right)^n$ $P_n\left(x\right)=f\left(c\right)+f'\left(c\right)\left(x-c\right)+\frac{f^n\left(c\right)}{2!}\left(x-c\right)^2+...+\frac{f^n\left(c\right)}{n!}\left(x-c\right)^n$ is the n^th Taylor polynomial for f at c.

Taylor series - A power series for an infinitely differentiable function throughout some interval containing c as an interior point and of the form $LaTeX: \sum_{n=0}^{\infty}\frac{f^{\left(n\right)}\left(c\right)}{n!}\left(x-c\right)^n$ $\sum_{n=0}^{\infty}\frac{f^{\left(n\right)}\left(c\right)}{n!}\left(x-c\right)^n$ , where f⁽ⁿ⁾(c) derivative of f at c.

Taylor's formula - If f has derivatives of all orders in an open interval I containing a, then for each positive integer n and for each x in I,

$LaTeX: f\left(x\right)=f\left(a\right)+f'\left(a\right)\left(x-a\right)+\frac{f^n\left(a\right)}{n!}\left(x-a\right)^n+R_n\left(x\right)$ $f\left(x\right)=f\left(a\right)+f'\left(a\right)\left(x-a\right)+\frac{f^n\left(a\right)}{n!}\left(x-a\right)^n+R_n\left(x\right)$

where $LaTeX: R_n\left(x\right)=\frac{f^{\left(n+1\right)}\left(c\right)}{\left(n+1\right)!}\left(x-a\right)^{n+1}$ $R_n\left(x\right)=\frac{f^{\left(n+1\right)}\left(c\right)}{\left(n+1\right)!}\left(x-a\right)^{n+1}$ for some c between a and x.

Taylor's theorem - If a function f and its first n derivatives f', f'', ... , f⁽ⁿ⁾

are continuous on the closed interval [a, b] and f is differentiable on the open interval (a, b), then there exists a number c between a and b such that

$LaTeX: f\left(b\right)=f\left(a\right)+f'\left(a\right)\left(b-a\right)+\frac{f''\left(a\right)}{2!}\left(b-a\right)^2+...+\frac{f^{\left(n\right)}\left(a\right)}{n!}\left(b-a\right)^{n+1}$ $f\left(b\right)=f\left(a\right)+f'\left(a\right)\left(b-a\right)+\frac{f''\left(a\right)}{2!}\left(b-a\right)^2+...+\frac{f^{\left(n\right)}\left(a\right)}{n!}\left(b-a\right)^{n+1}$ ,

where the remainder term $LaTeX: R_n\left(x\right)=\frac{f^{\left(n+1\right)}\left(c\right)}{\left(n+1\right)!}\left(b-a\right)^{n+1}$ $R_n\left(x\right)=\frac{f^{\left(n+1\right)}\left(c\right)}{\left(n+1\right)!}\left(b-a\right)^{n+1}$ .

Telescoping series - Any series where nearly every term cancels with a preceding or following term.

IMAGES CREATED BY GAVS