The sequence of numbers (or functions) in which the first (usually) number is given and every next number (term, element) is defined via prior number (or numbers) by an explicit fomula or a recurrent relationship. The sequence of natural numbers

1.2.3.4.5..........

is the example of an arithmetic series in which each next number can be found by adding 1 to the prior number, and the first number is 1. In the arithmetic series the difference between two consequitive elements is constant, such as

1 3 5 7 9 11 .... 5 15 25 35 45 ..........

In a geometric series the ratio of two consequitive elements is constant:

1 3 9 27 81 ........ 1/2 1/4 1/8 1/16 1/32 ........

The have a lot of applications in real life, physics, calculus. Some integrals in calculus can be calculated with the use of infinte arithmetic series. It helps to test integration formulas if you are doing this first time

(example: area under a parabola for the given domain, volume of a sphere, etc.). Geometric series can be applied to simplify algebraic relations (such as polynomials). They are also used in quantum mechanics, as well as other areas of natural sciences. This
list can be continued.