Chapter-7-EXPONENTIAL AND LOGARITHMIC SERIES-(E)-01-Theory
Exponential Series
7.1 Definition (The number e) .
The limiting value of when n tends to infinity is denoted by e
i.e., e = = 2.71 (Nearly)
7.2 Properties of e .
(1) e lies between 2.7 and 2.8. i.e., 2.7 < e < 2.8 (since )
(2) The value of e correct to 10 places of decimals is 2.7182818284
(3) e is an irrational (incommensurable) number
(4) e is the base of natural logarithm (Napier logarithm) i.e. and is known as Napierian constant. ,
7.3 Exponential Series .
For , or
The above series known as exponential series and is called exponential function. Exponential function is also denoted by exp. i.e. ;
7.4 Exponential Function ax, where a > 0 .
....(i) , where . We have,
Replacing x by in this series,
Hence from (i),
7.5 Some Important Results from Exponential Series .
We have the exponential series
(1) …..(i)
(2) Replacing x by –x in (i), we obtain …..(ii)
(3) Putting in (i) and (ii), we get,
(4) From (i) and (ii), we obtain
(5)
Note :
7.6 Some Standard results.
(1) (2)
(3) (4)
(5) (6)
(7)
(8)
(9)
General term in the expansion of and coefficient of in
(10)
General term in the expansion of and coefficient of in
(11)
General term in the expansion of and coefficient of in
(12) (13)
(14) (15)
Example: 1
(a) (b) e (c) 2e (d) 3e
Solution: (b)
Example: 2
(a) e (b) 2 e (c) (d) 1/e
Solution: (d) Here
Example: 3 =
(a) 2e (b) 3 e (c) 4 e (d) 5 e
Solution: (d)
Here
Example: 4 The coefficient of in the expansion of is
(a) (b) (c) (d)
Solution: (d) We have coefficient of in
Example: 5
(a) e (b) 3 e (c) e/2 (d) 3e/2
Solution: (d)
Logarithmic Series
7.7 Logarithmic Series.
An expansion for as a series of powers of x which is valid only when, ,
Expansion of if then
7.8 Some Important Results from the Logarithmic Series .
(1) Replacing by in the logarithmic series, we get
or
(2) (i)
(ii) or
(3) The series expansion of may fail to be valid if |x| is not less than 1. It can be proved that the logarithmic series is valid for x=1. Putting x=1 in the logarithmic series.
We get,
(4) When , the logarithmic series does not have a sum. This is in conformity with the fact that log(1-1) is not a finite quantity.
7.9 Difference between the Exponential and Logarithmic Series.
(1) In the exponential series all the terms carry positive signs whereas in the logarithmic series the terms are alternatively positive and negative.
(2) In the exponential series the denominator of the terms involve factorial of natural numbers. But in the logarithmic series the terms do not contain factorials.
(3) The exponential series is valid for all the values of x. The logarithmic series is valid when |x|< 1.
Example: 6
(a) (b) (c) (d)
Solution: (a) We know that,
Putting x = 0.5, we get,
Example: 7
(a) (b) (c) (d)
Solution: (a) We know that, ………(i)
Also ………(ii)
By adding (i) and (ii), we get,
Example: 8 The coefficient of in the expansion of is
(a) (b) (c) (d) None of these
Solution: (b) We have,
So coefficient of
Example: 9 The equation holds for
(a) (b) (c) (d)
Solution: (c) Given equation hold for
Example: 10 If , then []
(a) (b) (c) (d)
Solution: (c)
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