Exponential Series

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Session 1
Exponential Series Logarithmic Series
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Session Objectives

• The number e
• Exponential Series
• Logarithmic Series

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The number e
Let us consider the series
The sum of this series is denoted by e.
To prove that
Using binomial theorem
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The number e
The number e is an irrational number and its
value lies between 2 and 3.
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Exponential Series
We have
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Exponential Series
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Some Results
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Some Results
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Some Results
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Some Important Deductions
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Exponential Theorem
General term of eax
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Logarithmic Series
If x lt 1, then
Replacing x by x,
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Logarithmic Series
(i) (ii)
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Logarithmic Series
Putting x 1 in (i), we get
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Class Exercise - 1
Solution
Comparing the coefficients of like powers of n
fromboth sides, we get
A 0, B C 2D 0, C 3D 0, D 1
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Solution contd..
e 3e e 5e
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Class Exercise - 2
Solution
The given series can be written as
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Solution contd..
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Class Exercise - 3
Find the coefficient of Xn in theexpansion of
eex.
Solution
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Solution contd..
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Class Exercise - 4
Solution
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Solution contd..
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Class Exercise - 5
Solution
Now we will find the nth term of the numerator
Sn 4 11 22 37 ... tn 1 tn


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Subtracting, 0 4 7 11 15 19 ... (tn
tn 1) tn
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Solution contd..
2n2 n 1
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Solution contd..
2e 3e (e 1)
6e 1
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Class Exercise - 6
Solution
Let Tr be the nth term of the infinite series.
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Solution contd..
Comparing the coefficients of n2, n andconstant
term from both sides of theequation (ii), we
get4A 4B 4C 0, 2A 2C 2 and B
3Solving the above equations, we get A 2, B
3, C 1
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Solution contd..
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Solution contd..
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Class Exercise - 7
Solution
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Solution contd..
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Class Exercise - 8
Solution
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Solution contd..
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Class Exercise - 9
Solution
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Solution contd..
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Class Exercise - 10
Solution
By componendo and dividendo
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Solution contd..
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