# Exponential series

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Exponential series is a useful tool in various application of mathematics as well as wide application in physics also. The exponential series can also be expanded as infinite series in this way:-
Series: - e ^x = 1 + x + (x^2)/ 2! + (x^3)/3! + …..

Here! is factorial notation.

This series expansion helps to evaluate the complex problems in a very easy manner.  Only thing here is to remember the general expansion of the exponential series. This tool guides to solve problems where complex exponential function involves, and without solving this one cannot even think to solve such complicated problems. The above important points can be more clarified by the following examples:-

Question 1:- Expand exponential series e^1

Solution 1:- We know that the general expansion of exponential series is:-

Series: - e ^x = 1 + x + (x^2)/ 2! + (x^3)/3! + …..

Now by substituting x = 1 in above equation

Therefore e^1 = 1+ 1 + (1^2)/2! + (1^3)/3! +…

So e ^1 is approximately equals to 2.67

Question 2:- Expand exponential series e^2

Solution 2:-   We know that the general expansion of exponential series is:-

Series: - e ^x = 1 + x + (x^2)/ 2! + (x^3)/3! + …..

This implies, e ^2 = 1 + 2 + (4)/ 2 + (8)/3*2+ …..

Now by substituting x = 2 in above equation

Therefore e^1 = 1+ 2 + (2^2)/2! + (2^3)/3! +…

So e ^1 is approximately equals to 7.3