3 Sequences and series

These notes are reusing text and imagery from: Calculus Volume 1 from OpenStax, Print ISBN 193816802X, Digital ISBN 1947172131, https://www.openstax.org/details/calculus-volume-1 and Precalculus from OpenStax, Print ISBN 1938168348, Digital ISBN 1947172069, https://www.openstax.org/details/precalculus

3.1 Sequences

Definition: A sequence is a function whose domain is the set of positive integers. It is the ordered list of numbers \(\{a_1, a_2, a_3, \ldots, a_n, \ldots\}\).

Each number \(a_n\) in the sequence is called a term, the subscript \(n\) is called the index variable.

If the sequence continues infinitely, we call it infinite sequence, otherwise we call it finite.

We often denote the whole sequence as \(\{ a_n \}_{n=1}^{\infty}\) or \(\{ a_n \}\).

Sequences may be defined by

  • listing the terms
  • explicit formulas \(f(n) = a_n\), e.g. \(f(n) = 2^n\) defines the sequence \(1, 2, 4, 8, 16, \ldots\)
  • recurrence relations, e.g. \(a_1 = 2, a_n = 2 a_{n-1}\)

3.1.1 Arithmetic sequence

Definition: An arithmetic sequence is a sequence for which the difference \(d = a_n - a_{n-1}\) between any two consecutive terms is constant so that

  • listing the terms: \(\{a_n\} = \{a_1, a_1 + d, a_1 + 2d, a_1 + 3d, \ldots \}\).
  • explicit formula: \(a_n = cn + b\)
  • recursive relation: \(a_n = a_{n-1} + d\)

3.1.2 Geometric sequence

Definition: A geometric sequence is a sequence for which the ratio between any two consecutive terms \(r = \frac{a_n}{a_{n-1}}\) is constant so that

  • listing the terms: \(\{a_n\} = \{a_1, r a_1 r, r^2 a_1, r^3 a_1, \ldots \}\).
  • explicit formula: \(a_n = c r^{n}\)
  • recursive relation: \(a_n = r a_{n-1}\)

In figure, (a) is geometric sequence, the percentage change (ratio) is constant, (b) is arithmetic sequence, the rate of change is constant

3.2 Limit of a sequence

For an infinite sequence, explore the behaviour as \(n \to \infty\).

For example:

  • \(\{1 + 3n\} = \{ 4, 7, 10, 13, \ldots \}\) becomes arbitrarily large so that 1 + 3n as \(n \to \infty\).
  • \(\{1 - (\frac{1}{2})^n \} = \{ \frac{1}{2}, \frac{3}{4}, \frac{7}{8}, \frac{15}{16}, \ldots \}\) we have \(1 - (\frac{1}{2})^n \to 1\) as \(n \to \infty\).
  • \(\{ (-1)^n \} = \{ -1, 1, -1, 1, \ldots \}\) alternates and does not converge as \(n \to \infty\).

Definition: If the terms \(a_n\) of a sequence \(\{ a_n \}\) become arbitrarily close to a constant \(L\) as \(n\) becomes sufficiently large, we say the sequence converges with the limit \(L\). \[\lim_{n \to \infty} a_n = L\] If a sequence does not converge we say it diverges and the limit does not exist.

Formally: A sequence \(\{ a_n \}\) converges to a real number \(L\) if for all \(\epsilon > 0\), there exists an integer \(N\) such that for all \(n \geq N\) we have \(|a_n - L| < \epsilon\). \(L\) is the limit of the sequence.

For the examples above:

  • \(\lim_{n \to \infty}(1 + 3n) = \infty\) divergent.
  • \(\lim_{n \to \infty}(1 - (\frac{1}{2})^n) = 1\) convergent.
  • $_{n }( (-1)^n ) $ does not exist, divergent.


3.2.1 Limits of geometric sequence

For a geometric sequence \(\{r^n \}\)

\[\begin{eqnarray} r^n \to 0 & \text{ if } -1 < r < 1, & (\text{equivalently, if } |r| < 1)\\ r^n \to 1 & \text{ if } r = 1 & \\ r^n \to \infty & \text{ if } r > 1 & \\ r^n \quad & \text{diverges if } r \leq -1 & \\ \end{eqnarray}\]


3.2.2 Algebraic limit laws

\[\begin{eqnarray} \lim_{n \to \infty} c & = & c \\ \lim_{n \to \infty} c a_n & = & c \lim_{n \to \infty} a_n \\ \lim_{n \to \infty} (a_n \pm b_n) & = & \lim_{n \to \infty} a_n \pm \lim_{n \to \infty} b_n \\ \lim_{n \to \infty} (a_n \times b_n) & = & \big(\lim_{n \to \infty} a_n \big) \times \big(\lim_{n \to \infty} b_n \big) \\ \lim_{n \to \infty} \frac{a_n}{b_n} & = & \frac{\lim_{n \to \infty} a_n}{\lim_{n \to \infty} b_n} \text{ if } \lim_{n \to \infty} b_n \neq 0 \text{ and } b_n \neq 0 \text{ for all } n \end{eqnarray}\]


3.2.3 Bounded sequence

Definition: A sequence \(\{ a_n \}\) is bounded above if there exists a real number \(M\) such that \(a_n \leq M\) for all \(n\).
A sequence is bounded below if there exists a real number \(M\) such that \(a_n \geq M\) for all \(n\).
A sequence is bounded if it is bounded above and below. Otherwise it is called unbounded.

Note: If a sequence \(\{ a_n \}\) converges, it is bounded. (But not the other way round.)

For example:

  • \(\{ 1/n \}\) is bounded below and above because \(0 \leq 1/n \leq 1\) for all \(n\)
  • \(\{ 2^n \}\) is bounded below but not above because \(0 \leq 2^n\) but has no upper bound (\(\lim_{n \to \infty} 2^n = \infty\)).
  • \(\{ (-1)^n \}\) is bounded below and above because \(-1 \leq (-1)^n \leq 1\) for all \(n\). However, the series diverges!


3.2.4 Increasing and decreasing sequences

Definition: A sequence \(\{ a_n \}\) is increasing for \(n \geq n_0\) if \(a_{n+1} /geq a_n\), it is decreasing if \(a_{n+1} /leq a_n\). The sequence is monotonically increasing (decreasing) if it is increasing (decreasing) for all \(n \geq n_0\).

Note: If a sequence \(\{ a_n \}\) is bounded and monotone, it converges.

We say the sequence is strictly increasing (decreasing) if the consecutive terms are strictly greater \(>\) (smaller \(<\)).

3.3 Series

Sum notation

Basic rules for working with sum notations

\[\begin{eqnarray} \sum_{i=1}^n c a_i & = & c \sum_{i=1}^n a_i \\ \sum_{i=1}^n (a_i \pm b_i) & = & \sum_{i=1}^n a_i \pm \sum_{i=1}^n b_i \\ \sum_{i=1}^n c & = & nc \qquad (c \text{ is a constant not depending on } i) \\ \sum_{i=1}^n i & = & \frac{n(n+1)}{2} \end{eqnarray}\]

3.3.1 Series

Definition: An infinite series is the infinite sum of a sequence \[\sum_{n = 1}^\infty a_n = a_1 + a_2 + a_3 + \cdots \enspace .\]

How do we sum infinite number of terms?

Definition: A partial sum of a series is the finite sum of a sequence of the form \[S_k = \sum_{n = 1}^k a_n = a_1 + a_2 + a_3 + \cdots + a_k \enspace .\] From these we can construct a sequence of partial sums \(\{ S_k \} = \{ S_1, S_2, \ldots \}\).

Compare: definition of series and the sequence of partial sums \[\sum_{n = 1}^\infty a_n = \lim_{k \to \infty} S_k \quad \text{(if it exists).}\] The infinite series converges (it exists) if the sequence of partial sums converges (it has got a limit). Otherwise, the infinite series diverges (it does not exist).

Note: The index of a series does not have to start from \(n = 1\) but any value of \(n\).

From the standard rules for summations we have the following algebraic rules \[\begin{eqnarray} \sum_{n=1}^\infty c a_n & = & c \sum_{n=1}^\infty a_n \\ \sum_{n=1}^\infty (a_n \pm b_n) & = & \sum_{n=1}^\infty a_n \pm \sum_{n=1}^\infty b_n \\ \end{eqnarray}\]

3.3.2 Harmonic series

The harmonic series is defined as \[\sum_{n = 1}^\infty \frac{1}{n} = 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \cdots\] Does the series exist (converge)?

  1. Limit properties of the sequence \(\{ \frac{1}{n} \}\) \[ \lim_{n \to \infty} \frac{1}{n} = 0\] Does this imply the series converges? NO! Need to explore the sequence of partial sums!

  2. Limit properties of the sequence \(\{ S_k = \sum_{n=1}^k \frac{1}{n} = 1, 1.5, 1.833, 2.083, 2.283, \ldots\}\) \[\lim_{k \to \infty} S_k = \lim_{k \to \infty} \sum_{n=1}^k \frac{1}{n} = \infty\]

3.3.3 Geometric series

We can write any geometric series in the form \[\sum_{n = 1}^\infty a r^{n-1} = a + a r + a r^2 + a r^3 + \cdots \] The sequence of partial sums \(S_k\) has the terms \[S_k = \sum_{n = 1}^k a r^{n-1} = a + a r + a r^2 + a r^3 + \cdots + a r^{k-1}\] Fro \(a > 0\) \[ S_k = \begin{cases} a k & \text{if } r = 1 \\ \frac{a (1-r^k) }{(1-r)} & \text{if } r \neq 1 \end{cases} \] Limit properties of the sequence of partial sums of a geometric series \[\sum_{n = 1}^\infty a r^{n-1} = \lim_{k \to \infty} S_k \begin{cases} \frac{a}{1-r} & \text{if } |r| < 1 \\ \text{diverges} & \text{if } |r| \geq 1 \end{cases} \]

3.3.4 Arithmetic series

We can write any arithmetic series in the form \[\sum_{n = 1}^\infty a + d (n-1) = a + (a + d) + (a + 2d) + (a + 3d) + \cdots \] The sequence of partial sums \(S_k\) has the terms \[S_k = \sum_{n = 1}^k a + d (n-1) = a + (a + d) + (a + 2d) + (a + 3d) + \cdots + (a + (k-1)d) = k a + d \sum_{n = 1}^{k-1} k = \frac{k}{2} (a_1 + a_k)\]

Careful: All arithmetic series diverge!

3.4 Some special notation

3.4.1 Factorials

Definition: Factorials \(n!\) are products of the consecutive sequence of integers from \(1\) to \(n\) so that \[n! = n \times (n-1) \times (n-2) \times \cdot \times 2 \times 1 = \prod_{i = 1}^n i \enspace .\] Note: The product notation \(\prod_{i=1}^n\) is similar to the sum notation \(\sum_{i=1}^n\) but instead of giving the sum (adding up), it gives the product (multiplicatoin) of the terms.

For example:

  • \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\)
  • \(4! = 4 \times 3 \times 2 \times 1 = 4\)
  • \(1! = 1\)
  • \(1! = 0\) (special case)

Useful: \[n! = n \, (n-1)!\]

Factorials are used to compute the number of permutations, that is the number of ways a number of objects can be arranged into a sequence (ordered set).

\(P(n, r)\) indicates the number of permutations (possible orderings) of \(r\) objects selected from a total of \(n\) objects \[P(n, r) = \frac{n!}{(n-r)!} \enspace .\]

3.4.2 Binomial coefficient

The binomial coefficient \[{n \choose r} = \frac{n!}{r! (n-r)!} \] is the number of ways selecting \(r\) objects from \(n\) posibilities without caring about the order.

\(C(n, r)\) indicates the number of combinations (unordered sets) of \(r\) objects selected from a total of \(n\) objects \[C(n, r) = {n \choose r} = \frac{n!}{r! (n-r)!} \enspace .\]