INFINITE SERIES
A series is, informally speaking, the sum of the terms of a sequence. Finite sequences and series have defined first and last terms, whereas infinite sequences and series continue indefinitely.
The terms of the series are often produced according to a certain rule, such as by a formula, or by an algorithm. As there are an infinite number of terms, this notion is often called an infinite series. Unlike finite summations, infinite series need tools from mathematical analysis, and specifically the notion of limits, to be fully understood and manipulated.
Differential Equations are frequently solved by using infinite series. Fourier series, Fourier-Bessel series, etc. expansions involve infinite series. Transcendental functions (trigonometric, exponential, logarithmic, hyperbolic, etc.) can be expressed conveniently in terms of infinite series. Many problems that cannot be solved in terms of elementary (algebraic and transcendental) functions can also be solved in terms of infinite series.
So, in other words, infinite series occur so frequently in all types of engineering problems that the necessity of studying their convergence or divergence is very important. Unless a series employed in an investigation is convergent, it may lead to illogical conclusion.
Hence, it is essential that the students of engineering begin by acquiring an intelligent grasp of this subject.