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You CAN Ace Calculus

17calculus > infinite series > sequences

Calculus Main Topics

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Sequences

Infinite sequences are an important introduction to infinite series, since infinite series are built upon infinite sequences. Notice the difference.

Sequence Series A sequence is just a list of items separated by commas. With a series, we actually add up some (or all) terms of some sequence.

Sequences can be infinite (have an infinite number of terms) or finite (have a finite number of terms).

 Example 1: $$\{ 1, 2, 3, 4, 5 \}$$ Example 2: $$\{ 1, 4, 9, 16, 25, 36, . . . \}$$ This is a finite sequence since it has a finite number of elements, i.e. after the number $$5$$ there are no more numbers in the sequence and it stops. This is an infinite sequence since it has an infinite number of elements. The infinite part is denoted by the three periods at the end of the list.

We don't always list the elements of a sequence. If there is a logical pattern, we can write it more compact form as an equation. This idea is discussed in the next section on notation.

Notation

Sequences may be written in several different ways. These are the ones you will come across the most in calculus.
1. As a list of items; we showed this in the two examples above.
2. $$\displaystyle{\{a_n\}_{n=1}^{n=\infty}}$$ or more simply as $$\displaystyle{\{a_n\}_{1}^{\infty}}$$; in example 2 above, we would also include what $$a_n$$ was, i.e. $$a_n = n^2$$.
3. Similar to the last way, we could write the sequence in example 2 above as $$\displaystyle{\{n^2\}_{n=1}^{n=\infty}}$$; in this case, $$a_n$$ is implied to be $$a_n = n^2$$.
4. You also often see a sequence written without specifying $$n$$, i.e $$\{a_n\}$$ and n is implied as $$n = 1, 2, 3, . . . \infty$$
5. A sequence can also be built from itself, called a recursive sequence. In these types of sequences, the first few terms are given and the following terms are defined based on those first terms. For example, $$a_1 = 1, ~~ a_{n+1} = 2a_n$$.
6. Finally, you can describe a sequence with words. This is shown in practice problem C01 below. This is a very unusual way to do it, so you won't see it much.

The whole idea is to emphasize that a sequence is a LIST of elements. Each element stands alone and does not interact at all with any other element.

Here is quick introduction video clip that will get you started. It has a couple of good examples.

 PatrickJMT - Sequences

Arithmetic Sequence

One special type of sequence is an arithmetic sequence, which looks like $$\{ a_n \}$$ where $$a_n = c+nk$$, $$c$$ and $$k$$ are constants. An example is $$\{ 3, 12, 21, 30, . . . \}$$. Notice that in this example, we start with $$3$$ and add $$9$$ to get the next element.

Geometric Sequence

Another special type of sequence is a geometric sequence. This sequence looks like $$\{ a_n \} = \{ c, cr, cr^2, cr^3, . . . \}$$ where the terms are of the form $$a_n = cr^n$$ for $$n=0, 1, 2 . . .$$
$$c$$ is a constant
$$r$$ is a term that does not contain $$n$$.
An example is $$\{ 1, 3, 9, 27, 81, . . . \}$$ where we start with $$1$$ and multiply by $$3$$ to get the next element. Compare this to the arithmetic sequence where we add to get the next element.

Sequence Convergence and Divergence

Convergence of a sequence is defined as the limit of the terms approaching a specific real number. In equation terms, it looks like this. For a sequence $$\{a_n\}$$ where $$n=\{1, 2, 3, 4, . . . \}$$, we calculate $$\displaystyle{ \lim_{n \to \infty}{a_n} = L }$$

Convergence: If $$L$$ is a real number, then we say that the sequence $$\{a_n\}$$ converges to $$L$$.

Divergence: If $$L = \pm \infty$$ or the limit cannot be determined, i.e. it is undefined, then the sequence is said to diverge.

Note: The index $$n$$ can start at zero instead of one. This is what happened with the geometric sequence shown in the previous section. In actuality, the index can start at any number. However, for consistency, we usually start it at either zero or one.

Using L'Hôpital's Rule

When evaluating the limit $$\displaystyle{ \lim_{n \to \infty}{a_n} = L }$$, we sometimes need to use L'Hôpital's Rule. In the course of using L'Hôpital's Rule, we will need to take some derivatives. However, derivatives are only defined on continuous functions. And since a sequence is a set of discontinuous values, we can't directly use L'Hôpital's Rule. Fortunately, we have a theorem that will help us.

Theorem

If we can find a continuous function, call it $$g(x)$$ where $$g(n) = a_n$$ for all $$n$$, then $$\displaystyle{ \lim_{x \to \infty}{g(x)} = \lim_{n \to \infty}{a_n} }$$.

Finding a function $$g(x)$$ is not usually that hard, if $$a_n$$ is given in equation terms. Just replace $$n$$ with $$x$$.

I know this seems kind of picky but you need to get used to following the exact requirements of theorem. That is one reason we suggest that you read through and try to understand proofs in calculus. Okay, let's get down to some details and how to work with sequences. These next two videos will help you a lot.

This video is a bit long but well worth taking the time to watch. It contains a lot of detail about sequences and how to work with them and has plenty of good examples.

 Dr Chris Tisdell - Sequences (part 1)

This video is a continuation of the previous video and is not as long but it discusses the core of sequences, boundedness and convergence behaviour. You need to watch the previous video first to get the context before watching this one.

 Dr Chris Tisdell - Sequences (part 2)

Binomial Theorem

The binomial theorem is a way to expand a term in the form $$(a+b)^n$$ into a series. We do not cover this topic extensively at this time but here is a video to get you started with an explanation of the equation and an example.

 PatrickJMT - The Binomial Theorem
 Okay, time for some practice problems. After that, you will be ready to begin working with infinite series. infinite series →

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Practice Problems

Instructions - - Unless otherwise instructed, write the first few terms of these sequences and determine if they converge or diverge. If not specified, start the index at n=1.

 Level A - Basic

Practice A01

$$\displaystyle{ \left\{ \frac{2n}{6n-5} \right\} }$$

solution

Practice A02

$$\displaystyle{\left\{\frac{n+1}{n^2}\right\}_{n=1}^{n=\infty}}$$

solution

Practice A03

$$\displaystyle{\left\{\frac{3n^2-1}{10n+4n^2}\right\}_{n=2}^{n=\infty}}$$

solution

Practice A04

Find the limit of the sequence $$\{a_n\}$$ where $$\displaystyle{a_n=\frac{2n}{5n-3}}$$.

solution

Practice A05

Find the limit of the sequence $$\{a_n\}$$ where $$\displaystyle{a_n=\frac{n^2-n+7}{2n^3+n^2}}$$.

solution

Practice A06

Find the nth term of the sequence $$\{ 1, 4, 9, 16 . . . \}$$

solution

Practice A07

Find a formula for the general term $$a_n$$ of the sequence $$\displaystyle{ \left\{ \frac{1}{2}, -\frac{4}{3}, \frac{9}{4}, -\frac{16}{5}, \frac{25}{6}, . . . \right\} }$$

solution

Practice A08

$$\{n(n-1)\}$$

solution

Practice A09

$$\displaystyle{\left\{1+\left(-\frac{1}{2}\right)^n\right\}}$$

solution

Practice A10

Compute $$\displaystyle{\lim_{n\to\infty}{\frac{\ln(n)}{n}} }$$

solution

Practice A11

Compute $$\displaystyle{\lim_{n\to\infty}{\sqrt[n]{n}} }$$ using the result from the previous problem.

solution

 Level B - Intermediate

Practice B01

$$\displaystyle{\left\{\frac{(-1)^{n+1}}{3^n}\right\}_{n=0}^{n=\infty}}$$

solution

Practice B02

For the sequence $$\displaystyle{\left\{\frac{1}{2n+3}\right\}}$$ determine whether it is increasing, decreasing, monotonic and if it is bounded. Explain your reasoning.

solution

Practice B03

$$\displaystyle{\left\{\frac{e^{3n}}{n^2}\right\}}$$

solution

Practice B04

$$\displaystyle{\left\{\frac{2^n}{3^{n+1}}\right\}}$$

solution

Write down the first few terms of the sequence $$\displaystyle{\{a_n\}}$$ where $$a_n$$ is the nth digit of $$e$$ and discuss the convergence or divergence of the sequence.