Calculus: Sequences

Definition of Sequences

And now for something completely different!

A sequence is a list of numbers that follow a particular pattern as expressed by some formula. Each number in the sequence is called a term. Calculus is mainly concerned with infinite series, which (wait for it) have an infinite number of terms.

A sequence in is written as a list of the terms enclosed by brackets, like so: 

$$\left\{a_1, a_2, a_3, \ldots \right\}$$

The terms in a sequence are usually generated by some kind of function:

$$a_n = f(n)$$

By convention, the notation $a_n$ is used to denote the $n$th term in the sequence. A sequence can also be expressed by putting this function value inside the brackets, with the initial term and the maximum term are respectively subscripted and superscripted to the right. The following expressions all denote the same set:

$$ \begin{array}{ccc} \{1,2,4,8,16,\ldots\} & a_n = 2^n & \left\{ 2^n \right\}_{n=1}^{\infty} \end{array}$$


  1. Find expressions to represent the following sequences:

    1. $\left\{1,\dfrac{1}{2}, \dfrac{1}{3}, \dfrac{1}{4}, \dfrac{1}{5}, \ldots\right\}$
    2. $\left\{\dfrac{1}{4}, \dfrac{1}{9}, \dfrac{1}{16}, \dfrac{1}{25}, \ldots\right\}$
    3. $\left\{1,-2,3,-4,5, \ldots\right\}$
    4. $\left\{2, 1, \dfrac{2}{3}, \dfrac{1}{2}, \dfrac{2}{5}, \dfrac{1}{3} \ldots\right\}$
    1. This sequence is just the reciprocals of the integers: $\left\{\dfrac{1}{n}\right\}_{n=1}^{\infty}$
    2. This sequence is the squared reciprocals of the integers starting from 2: $\left\{\dfrac{1}{n^2}\right\}_{n=2}^{\infty}$
    3. In order to alternate between positive and negative, raise $-1$ to the appropriate power: $\left\{n\cdot (-1)^{n+1}\right\}_{n=1}^{\infty}$
    4. This one's tricky, but it's just 2 times the reciprocals of the integers: $\left\{\dfrac{2}{n}\right\}_{n=1}^{\infty}$
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  2. Write out the first 6 terms of the following sequences:

    1. $a_n = \left\{\dfrac{n+1}{n+2}\right\}_{n=1}^{\infty}$
    2. $a_n = \left\{\dfrac{(-1)^n}{n^2+1}\right\}_{n=1}^{\infty}$
    3. $a_n = \left\{\cos\left(\dfrac{n\pi}{4}\right)\right\}_{n=1}^{\infty}$
    1. $\left\{\dfrac{2}{3}, \dfrac{3}{4}, \dfrac{4}{6}, \dfrac{5}{6}, \dfrac{6}{7}, \dfrac{7}{8} \right\}$
    2. $\left\{\dfrac{-1}{2}, \dfrac{1}{5}, \dfrac{-1}{10}, \dfrac{1}{17}, \dfrac{-1}{26}, \dfrac{1}{37}\right\}$
    3. $\left\{\dfrac{1}{\sqrt{2}}, 0, \dfrac{-1}{\sqrt{2}}, -1, \dfrac{-1}{\sqrt{2}}, 0 \right\}$
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  3. Write a function for the Fibonacci sequence:


    Oh, and don't Google it, you scrub.

    The Fibonacci sequence is given by the following formula:

    $$\left\{ \begin{array}{ll} 0 & n=1 \\ 1 & n = 2 \\ a_{n-1} + a_{n-2} & n > 2 \end{array} \right\}_{n=1}^{\infty}$$

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  4. Find a formula for the following sequence that does not use summation notation:

    $$\{1, 11, 111, 1111, 11111, \ldots \}$$

    First, consider the sequence where each term has been multiplied by 9:

    $$\{9, 99, 999, 9999, 99999, \ldots \}$$

    The trick here may be easier to spot. The formula for this one is:

    $$a_n = \left\{10^{n}-1\right\}_{n=1}^{\infty}$$

    To solve for the original sequence, simply divide by 9 again:

    $$a_n = \left\{\dfrac{10^{n}-1}{9}\right\}_{n=1}^{\infty}$$

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  5. Find three ways to represent the following sequence:

    $$\left\{1, 2, 1, 2, 1, 2, \ldots\right\}$$

    $$a_n = 1.5 + 0.5 * (-1)^{n}$$

    $$a_n = \left\{ \begin{array}{ll} 1 & n \text{ is odd} \\ 2 & n \text{ is event} \end{array} \right.$$

    $$a_n = 1 + \left|\cos\left(\dfrac{\pi n}{2} \right)\right|$$

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