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Subject  >  Math  >  Calculus  >  Exponential and Logarithmic Series

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 ex.
Exponential Theorem
       If a > 0, then prove that
Graph of Exponential Function
 
       We see that as x increases, the value of ex also increases indefinitely. Also, as x decreases, the value of ex tends toward zero. The function ex 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 ex+y = exyy.
       v). loga mn = loga m + loga n.
       vi). loga m/n = loga m - loga n.

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Exponential and Logarithmic Series
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