Infinite Series

An series is an infinite sum, which we think of as the sum of the terms of a sequence, $a_1 + a_2 + a_3 + \ldots.$ We write a series using summation notation as

$\sum_{n=1}^\infty a_n=a_1 + a_2 + a_3 + \cdots.$ DO: Convince yourself that $\displaystyle\sum_{i=1}^\infty a_i=\sum_{k=1}^\infty a_k=\sum_{n=1}^\infty a_n=a_1+a_2+a_3+\cdots$ . The

$i,k$ , and

$n$ are just indices.
Notice:
a sequence is a list of infinitely many terms

$a_1,a_2,a_2\ldots$ , but
a series is a sum of infinitely many terms

$a_1 + a_2 + a_3 + \cdots$ .

Addition is something we do to finitely many numbers, not to infinitely many - how can we add infinitely many numbers? Consider a series $\displaystyle\sum_{i=1}^\infty a_i=a_1 + a_2 + a_3 + \cdots.$ Instead of worrying about adding infinitely many $a_i$ , we take (finite) partial sums $s_n$ , where $s_n$ is the sum of the first $n$ terms of the series.

$\begin{eqnarray*}s_1&=&a_1\\s_2&=&a_1+a_2\\s_3&=&a_1+a_2+a_3\\&\vdots&\\s_n&=&a_1+a_2+\cdots+a_{n-1}+a_n\\ \end{eqnarray*}$ A partial sum makes perfect sense as a sum of finitely many numbers. We can, of course, write each of these partial sums in summation notation; for example

$s_2=\sum_{i=1}^2a_i, \qquad s_5=\sum_{i=1}^5 a_i,\qquad s_n=\sum_{i=1}^n a_i, \quad\text{ etc.}$ Notice that as

$n$ gets larger and larger,

$\displaystyle s_n=\sum_{i=1}^n=a_1+a_2+\cdots+a_n$ gets closer and closer to the infinite sum

$\displaystyle \sum_{i=1}^\infty a_i$ .

Since we cannot add infinitely many things, we make sense of the series by taking a limit. We define the series as the limit of the partial sums as

$n$ goes to infinity.

Let

$\displaystyle\displaystyle\sum_{i=1}^\infty a_i$ be a series, and let

$\displaystyle s_n = \sum_{i=1}^n a_i$ be its $n^{th}$ partial sum.
We define

$\displaystyle\sum_{i=1}^\infty a_i = \lim_{n \to \infty} s_n.$

If this limit exists and is finite and equal to

$s$ , we say the series is convergent and that

$\displaystyle\sum_{i=1}^\infty a_i=s$ . Otherwise, we say the series is divergent.

Often, we can tell that a series converges, but cannot tell what value it converges to. For most of these modules on series, we will be sharpening our notions of convergence of series.

Warning: It may be confusing, but we often say

$\displaystyle \sum_{n=1}^\infty a_n = \lim_{n \to \infty} s_n$ , where the indicies

$n$ on the left and on the right are not the same. It is normal for us to define a series

$\sum a_n$ in terms of

$n$ and partial sums

$s_n$ in terms of

$n$ as well, but the

$n$ 's are used differently.

In the video, much of this will be discussed.

Note: There is an error at the 8:05 mark of the video. The voice-over is correct, but what is written is a little different. The correct values are

$s_4=-2$ ,

$s_5=3$ and

$s_6=-3$ . This doesn't affect the point being made about divergence.

Integration by Parts

Integrals of Trig Functions

Trig Substitutions

Partial Fractions

Strategies of Integration

Improper Integrals

Modeling with Differential Equations

Euler's Method and Direction Fields

Separable Equations

Models of Growth

Linear Equations

Parametrized Curves

Calculus with Parametrized Curves

Polar Coordinates

Areas and Lengths of Polar Curves

Conic sections

Conic Sections in Polar Coordinates

Infinite Sequences

Infinite Series

Integral Test

Comparison Tests

Convergence of Series with Negative Terms

The Ratio and Root Tests

Strategies for testing Series

Power Series

Representing Functions as Power Series

Taylor and Maclaurin Series

Applications of Taylor Polynomials

Functions of 2 and 3 variables

Partial Derivatives

Differentiability and the Chain Rule

Multiple Integrals

Iterated Integrals over Rectangles

Double Integrals over General Regions

Double integrals in polar coordinates

Multiple integrals in physics

Integrals in Probability and Statistics

Change of Variables

Introduction to Series