Sequences and notation

When we list numbers in a specific order, we are forming a sequence. The list $2, 4, 6, 8, 10, \dots$ is a sequence of even numbers; so is $1, 1, 2, 3, 5, 8, 13, \dots$ (the Fibonacci sequence). What distinguishes a sequence from a set is order: the third entry is the third entry, and you cannot swap it with the fifth without changing the sequence.

Definitions

A sequence is a function whose domain is the set of natural numbers (or some initial segment $\{1, 2, \dots, n\}$). Instead of writing $f(1), f(2), f(3), \dots$ we write $a_1, a_2, a_3, \dots$ The number $a_n$ is the $n$-th term or general term.

A finite sequence has finitely many terms: $a_1, a_2, \dots, a_n$. An infinite sequence continues indefinitely: $a_1, a_2, a_3, \dots$

Three ways to specify a sequence:

1. By formula: $a_n = 2n + 1$ gives $3, 5, 7, 9, \dots$
2. By recursion (each term depends on previous ones): $a_1 = 1$, $a_n = a_{n-1} + 2$.
3. By rule in words: "the $n$-th prime number".

A series is the sum of the terms of a sequence: $a_1 + a_2 + a_3 + \dots$ When the sequence is finite, the series is a finite sum. When it is infinite, the series may or may not have a meaningful value , convergence is a Class XII topic.

Sigma notation

The Greek letter $\Sigma$ ("sigma") condenses sums: $\sum_{k=1}^{n} a_k = a_1 + a_2 + \dots + a_n.$

The variable $k$ (the index) takes every integer from $1$ to $n$. The label below $\Sigma$ is the starting index; the label above is the ending index. Index names are interchangeable: $\sum_{k=1}^n a_k = \sum_{j=1}^n a_j$.

Some standard sums you should commit to memory immediately: $\sum_{k=1}^n 1 = n, \quad \sum_{k=1}^n k = \frac{n(n+1)}{2}, \quad \sum_{k=1}^n k^2 = \frac{n(n+1)(2n+1)}{6}, \quad \sum_{k=1}^n k^3 = \left(\frac{n(n+1)}{2}\right)^2.$

Properties of $\Sigma$

Sigma is linear: $\sum (c\, a_k) = c \sum a_k, \quad \sum (a_k + b_k) = \sum a_k + \sum b_k.$

But not multiplicative: $\sum (a_k b_k) \ne (\sum a_k)(\sum b_k)$ in general.

Worked examples

Example 1. Write the first five terms of $a_n = 3n - 2$.

$a_1 = 1, a_2 = 4, a_3 = 7, a_4 = 10, a_5 = 13$.

Example 2. Find the $n$-th term of $1, 4, 9, 16, 25, \dots$

These are $1^2, 2^2, 3^2, \dots$ so $a_n = n^2$.

Example 3. A sequence is defined by $a_1 = 2$, $a_{n+1} = 2 a_n + 1$. Find $a_4$.

$a_1 = 2$, $a_2 = 5$, $a_3 = 11$, $a_4 = 23$.

Example 4. Compute $\sum_{k=1}^{20} (2k - 1)$.

$\sum_{k=1}^{20} (2k - 1) = 2 \sum_{k=1}^{20} k - \sum_{k=1}^{20} 1 = 2 \cdot \dfrac{20 \cdot 21}{2} - 20 = 420 - 20 = 400$.

(This is the sum of the first $20$ odd numbers, which equals $20^2 = 400$.)

Example 5. Compute $\sum_{k=1}^{10} k(k + 1)$.

$\sum k(k+1) = \sum k^2 + \sum k = \dfrac{10 \cdot 11 \cdot 21}{6} + \dfrac{10 \cdot 11}{2} = 385 + 55 = 440$.

Try it yourself

1. Find the first five terms of $a_n = n^2 - n$.
2. Write the $n$-th term of $2, 5, 10, 17, 26, \dots$
3. A sequence: $a_1 = 1, a_2 = 1, a_n = a_{n-1} + a_{n-2}$. Find $a_7$.
4. Compute $\sum_{k=1}^{15} (3k + 2)$.
5. Compute $\sum_{k=1}^{12} k^2$.
6. Write in sigma notation: $1 + 4 + 7 + 10 + \dots + 31$.
7. Write the $n$-th term of $\dfrac{1}{2}, \dfrac{2}{3}, \dfrac{3}{4}, \dfrac{4}{5}, \dots$
8. Find $\sum_{k=1}^{n} (k^2 - k)$.
9. A sequence has $a_n = \dfrac{n}{n + 1}$. Find $a_{99}$.
10. Compute $\sum_{k=1}^{20} (k^2 + 2k)$.
11. Determine if $7, 10, 13, 16, \dots$ has $a_{100} = 304$. Verify.
12. Write the recursion that produces $3, 9, 27, 81, \dots$

Pitfalls / Tricks

• "Sequence" and "series" are distinct: the first is a list, the second is a sum.
• A recursive definition needs initial values plus a recurrence.
• $\sum (a_k b_k) \ne (\sum a_k)(\sum b_k)$ , never distribute multiplication over sigma like that.
• Insight. When you see an unfamiliar series, first ask which formula gives the general term? Once you have $a_k$, summation becomes algebra.