Other Series

Other Series

Natural Number Series

The numbers 1, 2, 3, 4, 5, 6 are called the natural numbers.

By adding the terms to form a series, 1 + 2 + 3 + 4 + 5 + 6 + ... n we have the sum of the natural numbers, which is written as:

Since this is an AP where the first term a = 1 and the common difference d = 1, we can evaluate the sum to be:

If we square the natural numbers and add the terms to obtain the series

12 + 22 + 32 + 42 + 52 + 62 + ... n2

For the sum of the square numbers, we have:

Finally, the sum of the cubes of the first n natural numbers is:

Recurrence relations

A recurrence relation is when there is a link between two or more of the terms of a sequence. The first few terms are usually given.

For example:

if we the first two terms of a sequence are T1 = 2 and T2 = 6 and we know that Tn+2 = Tn+1 + Tn, then we can find the next two terms as follows.

When n = 1, T3 = T2 + T1 = 6 + 2 = 8

When n = 2, T4 = T3+ T2 = 8 + 6 = 14

This gives us the sequence

2, 6, 8, 14, 22, 36,...

which is a variation on the Fibonacci sequence.

Periodicity

A sequence that repeats its terms in the same order after a certain number of terms is called periodic or cyclic. In trigonometry, we see how the sine and cosine graphs show a range between 1 and -1, and they repeat every 2π radians.

Therefore, the sequence T = cos(/2) has the terms 0, -1, 0, 1,... which repeat every four terms.

Oscillation

A sequence such as T = (-1)n has the terms -1, 1, -1, 1,...

This oscillates finitely between -1 and 1.

However the sequence T = (-10)n has the terms -10, 100, -1000, 10 000,...

This oscillates infinitely between −∞ and ∞.