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! + ...Ą 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 Ą, 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! + ...Ą 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.

Special Offer for NBC viewers!
		Get unlimited online tutoring for all subjects at $49.99 per month! (100% money-back guarantee)

	Introduction
	The number e
	Exponential Series
	Exponential Theorem
	Graph of Exponential Function
	Some Particular Exponential Series
	Graph of Logarithmic Series
	Logarithmic Series
	Some Particular Logarithmic Series
	Summary

All Subjects

	English
	Math
	Biology
	Chemistry
	Physics
	Science
	Statistics