# Other Series

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## Other 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

1^{2} + 2^{2} + 3^{2} + 4^{2} + 5^{2} + 6^{2} + ... n^{2}

**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 T_{1} = 2 and T_{2} = 6 and we know that T_{n+2} = T_{n+1} + T_{n}, then we can find the next two terms as follows.

When n = 1, T_{3} = T_{2} + T_{1} = 6 + 2 = 8

When n = 2, T_{4} = T_{3}+ T_{2} = 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(*nπ*/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 ∞.