

# What is a sequence? What is a limit?

A sequence is a collection of numbers that are listed in a specific order, typically with a common difference or ratio. The limit of a sequence refers to the value that the sequence approaches as the terms get infinitely close to each other.

Consider the sequence:

$$a_n = \frac{1}{n}$$

The limit of this sequence is 0.

The concept of a limit is fundamental in calculus and real analysis, as it allows us to study the behavior of functions and sequences as they approach specific values.

## Exercise
Consider the sequence:

$$b_n = \frac{1}{2^n}$$

Find the limit of this sequence.

### Solution
0

# Defining the limit of a sequence

To formally define the limit of a sequence, we use the following notation:

$$\lim_{n \to \infty} a_n = L$$

This means that the sequence $a_n$ converges to the limit $L$ as $n$ approaches infinity.

Consider the sequence:

$$a_n = \frac{1}{n}$$

We can formally define its limit as:

$$\lim_{n \to \infty} \frac{1}{n} = 0$$

To prove that a sequence converges, we must show that the terms of the sequence get infinitely close to a specific value. This is typically done by showing that the difference between the terms and the limit gets arbitrarily small.

## Exercise
Prove that the sequence defined by:

$$b_n = \frac{1}{n}$$

converges to 0.

### Solution
To prove that $\lim_{n \to \infty} \frac{1}{n} = 0$, we can use the following argument:

Let $\epsilon > 0$ be given. We want to find an $N$ such that for all $n > N$, we have $|\frac{1}{n} - 0| < \epsilon$.

Choose $N = \frac{1}{\epsilon}$. Then, for all $n > N$, we have:

$$|\frac{1}{n} - 0| = \frac{1}{n} < \frac{1}{N} = \epsilon$$

Therefore, $\lim_{n \to \infty} \frac{1}{n} = 0$.

# Examples of convergent and divergent sequences

Here are some examples of convergent and divergent sequences:

- Convergent sequence:

$$a_n = \frac{1}{n}$$

This sequence converges to 0.

- Divergent sequence:

$$b_n = \frac{1}{2^n}$$

This sequence does not converge to a finite limit.

- Convergent sequence:

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

This sequence converges to 0.

- Divergent sequence:

$$d_n = \frac{1}{n^2}$$

This sequence does not converge to a finite limit.

It is important to understand the behavior of sequences to determine their convergence or divergence.

## Exercise
Classify the following sequences as convergent or divergent:

- $e_n = \frac{1}{n^2}$
- $f_n = n$
- $g_n = \frac{1}{n}$

### Solution
- $e_n = \frac{1}{n^2}$ is convergent.
- $f_n = n$ is divergent.
- $g_n = \frac{1}{n}$ is convergent.

# Comparison tests for convergence and divergence

There are several comparison tests that can be used to determine the convergence or divergence of a sequence:

- Ratio test
- Root test
- Integral test
- Cauchy criterion

These tests are based on comparing the behavior of a sequence to that of another sequence whose convergence or divergence is known.

Consider the sequence:

$$a_n = \frac{1}{n}$$

We can use the ratio test to determine its convergence:

$$\lim_{n \to \infty} \frac{a_{n+1}}{a_n} = \lim_{n \to \infty} \frac{\frac{1}{n+1}}{\frac{1}{n}} = \lim_{n \to \infty} \frac{n}{n+1} = 1 < 1$$

Since the limit is less than 1, the sequence converges.

These tests are powerful tools for analyzing the behavior of sequences.

## Exercise
Apply the ratio test to the sequence:

$$b_n = \frac{1}{2^n}$$

### Solution
$$\lim_{n \to \infty} \frac{b_{n+1}}{b_n} = \lim_{n \to \infty} \frac{\frac{1}{2^{n+1}}}{\frac{1}{2^n}} = \lim_{n \to \infty} \frac{2^n}{2^{n+1}} = \frac{1}{2} < 1$$

Since the limit is less than 1, the sequence converges.

# The ratio test

The ratio test is used to determine the convergence or divergence of a sequence based on the ratio of consecutive terms:

$$\lim_{n \to \infty} \frac{a_{n+1}}{a_n}$$

If the limit is less than 1, the sequence converges. If the limit is greater than 1, the sequence diverges. If the limit is 1, the test is inconclusive.

Consider the sequence:

$$a_n = \frac{1}{n}$$

We can use the ratio test to determine its convergence:

$$\lim_{n \to \infty} \frac{a_{n+1}}{a_n} = \lim_{n \to \infty} \frac{\frac{1}{n+1}}{\frac{1}{n}} = \lim_{n \to \infty} \frac{n}{n+1} = 1 < 1$$

Since the limit is less than 1, the sequence converges.

The ratio test is a useful tool for analyzing the convergence of sequences.

## Exercise
Apply the ratio test to the sequence:

$$b_n = \frac{1}{2^n}$$

### Solution
$$\lim_{n \to \infty} \frac{b_{n+1}}{b_n} = \lim_{n \to \infty} \frac{\frac{1}{2^{n+1}}}{\frac{1}{2^n}} = \lim_{n \to \infty} \frac{2^n}{2^{n+1}} = \frac{1}{2} < 1$$

Since the limit is less than 1, the sequence converges.

# The root test

The root test is used to determine the convergence or divergence of a sequence based on the $n$-th root of the absolute value of the terms:

$$\lim_{n \to \infty} \sqrt[n]{|a_n|}$$

If the limit is less than 1, the sequence converges. If the limit is greater than 1, the sequence diverges. If the limit is 1, the test is inconclusive.

Consider the sequence:

$$a_n = \frac{1}{n}$$

We can use the root test to determine its convergence:

$$\lim_{n \to \infty} \sqrt[n]{\frac{1}{n}} = \lim_{n \to \infty} \frac{1}{\sqrt[n]{n}} = \lim_{n \to \infty} \frac{1}{n} = 0 < 1$$

Since the limit is less than 1, the sequence converges.

The root test is a useful tool for analyzing the convergence of sequences.

## Exercise
Apply the root test to the sequence:

$$b_n = \frac{1}{2^n}$$

### Solution
$$\lim_{n \to \infty} \sqrt[n]{\frac{1}{2^n}} = \lim_{n \to \infty} \frac{1}{\sqrt[n]{2^n}} = \lim_{n \to \infty} \frac{1}{2} = \frac{1}{2} < 1$$

Since the limit is less than 1, the sequence converges.

# The integral test

The integral test is used to determine the convergence or divergence of a sequence based on the integral of the terms:

$$\int_1^{\infty} |a_n|$$

If the integral converges, the sequence converges. If the integral diverges, the sequence diverges.

Consider the sequence:

$$a_n = \frac{1}{n^2}$$

We can use the integral test to determine its convergence:

$$\int_1^{\infty} \frac{1}{x^2} = \frac{1}{x}$$

Since the integral converges, the sequence converges.

The integral test is a powerful tool for analyzing the convergence of sequences.

## Exercise
Apply the integral test to the sequence:

$$b_n = \frac{1}{2^n}$$

### Solution
$$\int_1^{\infty} \frac{1}{2^n} = \frac{1}{\ln(2)}$$

Since the integral converges, the sequence converges.

# The Cauchy criterion

The Cauchy criterion is used to determine the convergence or divergence of a sequence based on the behavior of the terms as $n$ approaches infinity:

$$\lim_{n \to \infty} \frac{a_{n+p} - a_n}{a_n}$$

If the limit is 0 for all $p$, the sequence converges. If the limit is not 0 for some $p$, the sequence diverges.

Consider the sequence:

$$a_n = \frac{1}{n}$$

We can use the Cauchy criterion to determine its convergence:

$$\lim_{n \to \infty} \frac{a_{n+p} - a_n}{a_n} = \lim_{n \to \infty} \frac{\frac{1}{n+p} - \frac{1}{n}}{\frac{1}{n}} = \lim_{n \to \infty} \frac{1}{n(n+p)} = 0$$

Since the limit is 0, the sequence converges.

The Cauchy criterion is a useful tool for analyzing the convergence of sequences.

## Exercise
Apply the Cauchy criterion to the sequence:

$$b_n = \frac{1}{2^n}$$

### Solution
$$\lim_{n \to \infty} \frac{b_{n+p} - b_n}{b_n} = \lim_{n \to \infty} \frac{\frac{1}{2^{n+p}} - \frac{1}{2^n}}{\frac{1}{2^n}} = \lim_{n \to \infty} \frac{2^n(2^p - 1)}{2^{n+p}} = 0$$

Since the limit is 0, the sequence converges.

# Applications of convergence and divergence

The concepts of convergence and divergence have numerous applications in calculus and real analysis:

- Studying the behavior of functions and their limits
- Analyzing the convergence of series
- Developing numerical methods for solving differential equations
- Investigating the properties of geometric and fractal shapes

Consider the sequence:

$$a_n = \frac{1}{n}$$

This sequence converges to 0, which has important implications in calculus and real analysis.

Understanding the convergence and divergence of sequences is essential for solving complex problems in mathematics and engineering.

## Exercise
Discuss the applications of convergence and divergence in the following fields:

- Mathematical analysis
- Differential equations
- Numerical methods
- Geometry and fractal shapes

### Solution
- Mathematical analysis: Convergence and divergence are fundamental concepts in mathematical analysis, which helps in understanding the behavior of functions and their limits.

- Differential equations: The convergence and divergence of sequences are crucial in studying the behavior of solutions to differential equations, which are used in modeling various physical systems.

- Numerical methods: Convergence and divergence of sequences are important in developing numerical methods for solving differential equations, which are used in scientific computing and engineering.

- Geometry and fractal shapes: The concepts of convergence and divergence are used in understanding the properties of geometric and fractal shapes, which have wide-ranging applications in computer graphics, image processing, and pattern recognition.

# Summation of convergent and divergent series

The summation of a convergent series is the sum of its terms as $n$ approaches infinity:

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

The summation of a divergent series is undefined, as the sum does not approach a specific value.

Consider the convergent series:

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

Its summation is $\frac{\pi^2}{6}$.

The summation of convergent and divergent series is an important topic in calculus and real analysis.

## Exercise
Sum the following convergent series:

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

### Solution
$$\sum_{n=1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6}$$

# Limit laws and properties

Limit laws and properties are fundamental principles that govern the behavior of limits in calculus and real analysis:

- The limit of a constant is the constant itself.
- The limit of a sum is the sum of the limits.
- The limit of a product is the product of the limits.
- The limit of a quotient is the quotient of the limits.
- The limit of a composition of functions is the composition of the limits.

Consider the limit:

$$\lim_{x \to 0} x^2$$

By the power law for limits, we have:

$$\lim_{x \to 0} x^2 = 0$$

Limit laws and properties are essential for solving problems in calculus and real analysis.

## Exercise
Apply the limit laws and properties to the following limits:

- $\lim_{x \to 0} x^2$
- $\lim_{x \to 0} \frac{1}{x}$
- $\lim_{x \to 0} \frac{x^2 + 1}{x}$

### Solution
- $\lim_{x \to 0} x^2 = 0$
- $\lim_{x \to 0} \frac{1}{x}$ is undefined.
- $\lim_{x \to 0} \frac{x^2 + 1}{x} = 1$