# 8: Sequences and Series

- Page ID
- 4347

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)

- 8.1: Sequences
- A sequence is a list of objects in a specified order. We will typically work with sequences of real numbers and can also think of a sequence as a function from the positive integers to the set of real numbers. A sequence diverges if it does not converge.

- 8.2: Geometric Series
- Many important sequences are generated through the process of addition.

- 8.3: Series of Real Numbers
- An infinite series is a sum of the elements in an infinite sequence. The sequence of partial sums of a series P∞ k=1 ak tells us about the convergence or divergence of the series. The series converges if the sequence of partial sums converges.

- 8.4: Alternating Series
- An alternating series is a series whose terms alternate in sign. An alternating series converges if and only if its sequence of partial sums converges. The sequence of partial sums of a convergent alternating series oscillates around and converge to the sum of the series if the sequence of nth terms converges to 0.

- 8.5: Taylor Polynomials and Taylor Series
- We can use Taylor polynomials to approximate complicated functions. This allows us to approximate values of complicated functions using only addition, subtraction, multiplication, and division of real numbers. The Lagrange Error Bound shows us how to determine the accuracy in using a Taylor polynomial to approximate a function.

- 8.6: Power Series
- We can often assume a solution to a given problem can be written as a power series, then use the information in the problem to determine the coefficients in the power series. This method allows us to approximate solutions to certain problems using partial sums of the power series; that is, we can find approximate solutions that are polynomials. The connection between power series and Taylor series is that they are essentially the same thing.

- 8.E: Sequences and Series (Exercises)
- These are homework exercises to accompany Chapter 8 of Boelkins et al. "Active Calculus" Textmap.

## Contributors and Attributions

Matt Boelkins (Grand Valley State University), David Austin (Grand Valley State University), Steve Schlicker (Grand Valley State University)