Sequences and Series

Via repeated application of the same arithmetic operations on an initial number A one creates a logically connected sequence of numbers, that show interesting properties (to

Sequence guess the formation law of a sequence and thus to continue the initial numbers of a given sequence is popular type of riddle).

The following the letters

m, n, i, j

are used to indicate the position of terms in sequences. They can be

0

or positive integers.

If there is no upper limit for the number of terms in a sequence or for the terms in a series (

m \to \infty

), we refer to an infinite sequence or series.

4.1.1 Sequence and Series of the Natural Numbers

The especially simple arithmetic sequence of the natural numbers is created via the repeated addition of the unit 1; the individual term is characterized via the lower index (

1, 2, \dots in general m

), which itself is an increasing natural number.

\begin{matrix} A_{1} = 1; A_{n + 1} = A_{n} + 1 for n \geq 1 \to \\ A_{n} = 1, 2, 3, 4, 5, 6, . . . \end{matrix}

We now define the difference quotient for the terms of an arbitrary sequence with different indices

i

and

j

. This number is a measure for the change between two terms with different indices and thus for the growth of the sequence in the interval given by the indices:

\begin{matrix} Δ A_{i, j} = A_{i} - A_{j}; Δ_{i, j} = i - j \\ difference quotient: (\frac{Δ A_{i, j}}{Δ_{i, j}}) = \frac{A_{i} - A_{j}}{i - j} \end{matrix}

For consecutive terms the index interval is

1

and the difference quotient is equal to the difference of the terms:

\begin{matrix} Δ A_{i, i - 1} = A_{i} - A_{i - 1}; Δ_{i, i - 1} = i - (i - 1) = 1 \\ difference quotient: (\frac{Δ A_{i, i - 1}}{Δ_{i, i - 1}}) = A_{i} - A_{i - 1} \end{matrix}

For the sequence of the natural numbers the difference of consecutive terms is constant and equal to

1

. Therefore the difference quotient is also constant and equal to

1

From the terms of a sequence one can via repeated addition obtain a logically connected series, whose partial sums are again terms of a sequence. For the above example this would be the arithmetic series S and the sequence of its partial sums

\begin{matrix} S = 1 + 2 + 3 + 4 + 5 + 6 . . . \\ S_{1} = 1; S_{2} = 3; S_{3} = 6; S_{4} = 10; S_{5} = 15; S_{6} = 21 . . . \\ S_{n =} \sum_{m = 1}^{n} A_{n} = A_{1} + A_{2} + A_{3} + . . . + A_{n} \end{matrix}

For the sum sign

Σ

(capital Greek letter Sigma (S)) the index

m

of the sequence terms

A_{m}

runs from the number on the bottom to the number on top.

For the arithmetic series one can calculate the partial sums very easily from the indices. This rule is supposed to have been discovered by Gauß, when he had to sum up the numbers from

1

100

. This rule is founded on the symmetry of the series: two numbers that are symmetrically positioned relative to the middle of the partial sum add up always to the same sum

(n + 1)

and there are

n ∕ 2

such pairs.

The sequence of the natural numbers does not have an upper limit. The sum over its subsets increases faster with growing index in quadratic dependence on the index

4.1.2 Geometric Series

As another example we consider the sequence of powers of the real number

a

and the geometric series that is created from it via addition.

\begin{matrix} A_{0} = 1; A_{1} = a; A_{2} = a^{2}; A_{3} = a^{3}; A_{4} = a^{4} . . . \\ A_{n} = a^{n} for n \geq 0 \\ definition: Δ A_{i, j} = A_{i} - A_{j}; Δ_{i, j} = i - j \\ Δ A_{i, j} = a^{i} - a^{j} \\ {(\frac{Δ A}{Δ})}_{i, j} = \frac{a^{i} - a^{j}}{i - j}; {(\frac{Δ A}{Δ})}_{i, i - 1} = \frac{a^{i} - a^{(i - 1)}}{1} = a^{(i - 1)} (a - 1) \\ S_{n} = \sum_{m = 0}^{n} a^{m} = 1 + a + a^{2} + a^{3} + . . . a^{n} \end{matrix}

For the special case of

a = 1

the partial sums of the geometric series become an arithmetic sequence.

For

a

different from

1

the difference quotient depends on the index. For

a < 1

it keeps getting smaller; the terms of the sequence decrease, and the partial sums increase ever slower. For

a > 1

the difference quotient is positive and grows with the index; the terms of the sequence increase faster and faster, and the partial sums of the series even more.

4.1 Sequences and Series

4.1.1 Sequence and Series of the Natural Numbers

4.1.2 Geometric Series