Unit – 1C
Series
Q1) 
n
A1) L= 
= 
1/n
L= 
n =
= 
<1
Hence the series converges.
Q2) Find the Taylor series for the following:
= 
<1
A2)
(x/10) <1 and (x/10) > -1
Therefore, radius of convergence is (-10,10)
ROC =10
Q3) (a): f(n)5 = 
(b): f(x)=
A3)
(a): f(n)5 = 
Here the ROC 4
(b): f(x)=
= f (0) +f’ (0)x +
x2 + 
x3 +......
= 1+x+x2 +x3 + .....
= 
Q4) 
A4) Given
Q5) Leonhard Euler, the great Swiss mathematician introduced and named the number e in his calculus text in 1748.
A5) Given
Putting x=1 in there, we get:
Which gives
So, e is approximately 2.67 or 2<e<3 and the value of 
by using exp. Series
Which simplifies to:
To write 
as a series, it is best to rewrite 
as 
:
By adding & subtracting above two series, we get
Q6) Find the Fourier series of f(x) = x in the interval 
A6) Here 
; 
Its Fourier series is given by
… (1)
Where
&
Hence the required Fourier series is
Q7) Find a Fourier series expansion in the interval 
for
; 
; 
A7) Here
; 
; 
Hence, its Fourier series expansion is,
… (1)
Where
And
Hence equation (1) becomes
Q8) Find a Fourier series for
; 
A8)
Here
; 
Since f(x) is even function hence 
It’s Fourier series is
… (1)
Where
Hence equation (1) becomes,
Q9) Find half range cosine series of 
in the interval 
and hence deduce that
a) 
b) 
A9)
Here
; 
Hence it’s half range cosine series is,
… (1)
Where
Hence equation (1) becomes,
… (2)
Put x = 0, we get
Hence the result
Put 
we get,
i.e. 
Q10) Using complex form, find the Fourier series of the function
f(x) = sinnx = 
A10) We calculate the coefficients 
= 
=
Hence the Fourier series of the function in complex form is
We can transform the series and write it in the real form by renaming as
n=2k-1,n=
= 
Q11) Using complex form find the Fourier series of the function f(x) = x2, defined on the interval [-1,1]
A11)
Here the half-period is L=1. Therefore, the co-efficient c0 is,
For n
Integrating by parts twice, we obtain
= 
= 
= 
.
= 
.
Q12) Consider 
, 
A12) The Fourier expansion is,
By Parseval’s formulae
is Reiman Zeta function defined by:
