Exponential and Logarithmic Series

Introduction

     In this chapter, we shall study two series known as the Exponential series and Logarithmic series. In our discussion, we shall make use of mathematical tools like formula for sum of an infinite G.P., combinatorial coefficients.

The number e

     The sum of the infinite series 1 + 1/1! + 1/2! + 1/3! + 1/4! + ...¥ is called the exponential number and is denoted by 'e'.

Exponential Series

     If x is any complex number then the series is called the exponential series. It can be proved mathematically that this exponential series has a sum and we denote it by e^x.

Exponential Theorem

     If a > 0, then prove that

Graph of Exponential Function

     We see that as x increases, the value of e^x also increases indefinitely. Also, as x decreases, the value of e^x tends toward zero. The function e^x is one-one.

Some Particular Exponential Series

     The following are the Some Particular Exponential Series:

Graph of Logarithmic Series

 

     We see that as x increases from 0 to ¥, the value of log x also increases indefinitely. The function log x is one-one.

Logarithmic Series

      If x is a real number such that |x|<1, then the series is called the logarithmic series. It can be proved mathematically that this logarithmic series has the sum equal to log(1 + x).

Some Particular Logarithmic Series

     The following are the Some Particular Logarithmic Series:

Summary

     i). The sum of the infinite series 1 + 1/1! + 1/2! + 1/3! + 1/4! + ...¥ is called the exponential number.

     ii). 2 < e < 3.

     iii). The value of e rounded off to four decimal places is 2.7183.

     vi). For complex numbers x,y, we have e^x+y = e^xy^y.

     v). log_a mn = log_a m + log_a n.

     vi). log_a m/n = log_a m - log_a n.