

# Defining series convergence and divergence

Before diving into the various tests for testing series convergence, it's important to understand what convergence and divergence mean. A series is a sequence of numbers that we can add up. When the sum of a series converges, it means that the sum of the series gets arbitrarily close to a specific number as we add up more and more terms. On the other hand, when a series diverges, it means that the sum of the series gets arbitrarily large, or that it doesn't converge at all.

To formally define convergence and divergence, we use two mathematical concepts: limits and the $\epsilon$-$\delta$ definition of a limit. A series converges if its sum converges, and it diverges if its sum diverges.

For example, consider the geometric series:

$$\sum_{n=0}^{\infty} r^n$$

This series converges if $|r| < 1$, and diverges if $|r| > 1$.

# Geometric series convergence and divergence

A geometric series is a series where each term is a constant multiplied by the previous term. The general formula for a geometric series is:

$$\sum_{n=0}^{\infty} ar^n$$

The series converges if $|r| < 1$, and diverges if $|r| > 1$.

To see why, let's consider the partial sum of a geometric series:

$$S_n = \sum_{k=0}^{n} ar^k$$

If $|r| < 1$, then the terms of the series get smaller as $n$ increases, and the sum converges. If $|r| > 1$, then the terms of the series get larger as $n$ increases, and the sum diverges.

# Other types of series convergence and divergence

There are many other types of series, such as arithmetic series, harmonic series, and power series. The convergence and divergence of these series can be determined using various tests.

For example, the harmonic series:

$$\sum_{n=1}^{\infty} \frac{1}{n}$$

# Testing series convergence using the ratio test

The ratio test is used to determine the convergence or divergence of a series. The test is based on the ratio of consecutive terms in the series.

Let $a_n$ be the $n$-th term of the series. The ratio test states that if the limit of the absolute value of the ratio of consecutive terms exists and is less than 1, then the series converges. If the limit is greater than 1, the series diverges.

For example, consider the series:

$$\sum_{n=1}^{\infty} \frac{1}{n^2}$$

The ratio of consecutive terms is:

$$\frac{1}{(n+1)^2} \div \frac{1}{n^2} = \frac{n^2}{(n+1)^2}$$

As $n$ increases, the ratio approaches 1, so the series converges.

# Testing series convergence using the root test

The root test is another test for determining the convergence or divergence of a series. It is based on the n-th root of the absolute value of the $n$-th term in the series.

If the limit of the n-th root of the absolute value of the $n$-th term exists and is less than 1, then the series converges. If the limit is greater than 1, the series diverges.

For example, consider the series:

$$\sum_{n=1}^{\infty} \frac{1}{n!}$$

The n-th root of the absolute value of the $n$-th term is:

$$\frac{1}{n!}^{1/n} = \frac{1}{n}$$

As $n$ increases, the n-th root approaches 0, so the series converges.

# Testing series convergence using the integral test

The integral test is used to determine the convergence or divergence of a series based on the integral of the function that generates the series.

Let $f(x)$ be a function that generates the series. If the integral of $f(x)$ from 1 to infinity converges, then the series converges. If the integral diverges, then the series diverges.

For example, consider the series:

$$\sum_{n=1}^{\infty} \frac{1}{n^2}$$

The function that generates this series is $f(x) = \frac{1}{x^2}$. The integral of $f(x)$ from 1 to infinity converges, so the series converges.

# Testing series convergence using the comparison test

The comparison test is used to determine the convergence or divergence of a series by comparing it to another known series.

If the sum of the absolute values of the terms of the series is less than or equal to the sum of the absolute values of the terms of another known series, and the other series converges or diverges, then the original series converges or diverges as well.

For example, consider the series:

$$\sum_{n=1}^{\infty} \frac{1}{n}$$

We know that the harmonic series diverges. The terms of the harmonic series are less than or equal to the terms of our series, so our series diverges as well.

# The Cauchy convergence test

The Cauchy convergence test is a test for determining the convergence or divergence of a series based on the absolute values of the terms of the series.

If the absolute values of the terms of the series get smaller as $n$ increases, then the series converges. If the absolute values of the terms of the series do not get smaller, then the series diverges.

For example, consider the series:

$$\sum_{n=1}^{\infty} \frac{1}{n^2}$$

The absolute values of the terms of the series get smaller as $n$ increases, so the series converges.

# The Alternating Series Test

The Alternating Series Test is used to determine the convergence or divergence of a series based on the alternating signs of the terms of the series.

If the terms of the series alternate in sign and the absolute values of the terms of the series decrease monotonically, then the series converges. If the terms of the series do not alternate in sign or the absolute values of the terms of the series do not decrease monotonically, then the series diverges.

For example, consider the series:

$$\sum_{n=1}^{\infty} \frac{(-1)^n}{n}$$

The terms of the series alternate in sign and the absolute values of the terms of the series decrease monotonically, so the series converges.

# Taylor series convergence and divergence

Taylor series are a way to approximate a function as a power series. The convergence and divergence of a Taylor series depends on the function being approximated.

For example, consider the Taylor series for the function $e^x$:

$$\sum_{n=0}^{\infty} \frac{x^n}{n!}$$

This series converges for all values of $x$.

# Examples and exercises

In this textbook, we have covered various tests for testing series convergence and divergence. Here are some examples and exercises to help you practice your understanding:

## Exercise
1. Determine the convergence or divergence of the geometric series:

$$\sum_{n=0}^{\infty} \frac{1}{2^n}$$

2. Determine the convergence or divergence of the series:

$$\sum_{n=1}^{\infty} \frac{1}{n}$$

3. Determine the convergence or divergence of the series:

$$\sum_{n=1}^{\infty} \frac{(-1)^n}{n}$$

4. Determine the convergence or divergence of the Taylor series for the function $e^x$:

$$\sum_{n=0}^{\infty} \frac{x^n}{n!}$$

The answers to these exercises are:

1. The series converges because $|r| = \frac{1}{2} < 1$.

2. The series diverges because the terms of the series do not get smaller as $n$ increases.

3. The series converges because the terms of the series alternate in sign and the absolute values of the terms of the series decrease monotonically.

4. The series converges for all values of $x$ because the terms of the series decrease monotonically and the absolute values of the terms of the series get smaller as $n$ increases.