UNIT–3
Sequence and series
Sequence –
A function f : N 
, where S is a non-empty set , is called sequence , for each nϵN.
The sequence is written as f(1) , f(2) , f(3) , f(4)………. f(n).
Any sequence f(n) can be denoted as <f(n)> or {f(n)} or (f(n)).
Suppose f(n) = 
Then it can be written as - 
and can be denoted as <
> or {
} or (
)
is the n’th term of the sequence.
Example: suppose we have a sequence – 1 , 4 , 9 , 16 ,……….. And its n’th term is 
This sequence can be written as - <
>
Example: 
its n’th term will be 
.and can be written as <
>
Types of sequences –
1. Finite sequence- A sequence which has finite number of terms is called finite sequence.
2. Infinite sequence- A sequence which is not finite , called infinite sequence.
Limit of a Sequence- A sequence <
> is said to tend to limit “l” , when given any positive number ‘ϵ’ , however small , we can always find a integer ‘m’ such that | 
– l| < ϵ , for every for all, n≥m , and we can define this as follows,
Example: If 
, then the limit of 
will be,
= 
= 
= ½
Hence the limit of the sequence is 1/2 .
Some important limits to remember-
Convergent sequence- A sequence Sn is said to be convergent when it tends to a finite limit. That means the limit of a sequence Sn will be always finite in case of convergent sequence.
Divergent sequence- when a sequence tends to ±∞ then it is called divergent sequence.
Oscillatory sequence- when a sequence neither converges nor diverges then it is called oscillatory sequence.
Note- a sequence which neither converges nor diverges , is called oscillatory sequence.
A sequence converges to zero Is called null.
Example-1: consider a sequence 2, 3/2 , 4/3 , 5/4, …….. Here Sn = 1 + 1/n
Sol. As we can see that the sequence Sn is convergent and has limit 1.
According to def.
Example-2: consider a sequence Sn = n² + (-1)ⁿ.
Sol. Here we can see that, the sequence Sn is divergent as it has infinite limit.
Example: suppose 
, here the sequence is said to be oscillate. Because it is between -2 and 2.
Infinite series- If 
is a sequence , then 
is called the infinite series.
It is denoted by 
.
Examples of infinite series-
Covergent series - suppose n→∞ , Sn → a finite limit ‘s’ , then the seires Sn is said to be convergent .
We can denote it as,
Divergent series - when Sn tends to infinity then the series is said to be divergent.
Oscillatory series- when Sn does not tends to a unique limit (finite or infinite) , then it is called Oscillatory series.
Properties of infinite series –
1. The convergence and divergence of an infinite series is unchanged addition od deletion of a finite number of term from it.
2. If positive terms of convergent series change their sign , then the series will be convergent.
3. Let 
converges to s , let k be a non-zero fixed number then 
converges to ks.
4. Let 
converges to ‘l’ and 
converges to ‘m’.
Example-1: check whether the series 
is convergent or divergent. Find its value in case of convergent.
Sol. As we know that,
Sn = 
Therefore,
Sn = 
Now find out the limit of the sequence,
= ∞
Here the value of the limit is infinity, so that the series is divergent as sequence diverges.
Example-2: check whether the series 
is convergent or divergent. Find its value in case of convergent.
Sol. The general formula for this series is given by,
Sn = 
= 
)
We get,
) = 3/2
Hence the series is convergent and its values is 3/2.
Example-3: check whether the series 
is convergent or divergent.
Sol. The general formula can be written as,
We get on applying limits,
) = 3/4
This is the convergent series and its value is 3 / 4
Example-4: check whether the following series is convergent or divergent. If convergent , find its value.
Sol. n’th term of the series will be,
] = ½
Let 
be a series of positive terms and let
Then 
is convergent when l<1 and diverges when l >1.
Proof: case-1:
Or
Since,
Is a geometric series with common ratio < 1 so that the series will be convergent.
Case- 2:
By the limit concept, we can find a number,
That means
After 1st ‘r’ terms , each term is > 1
So that the series is divergent.
Example: Test the convergence of the series whose nth term is given below-
Sol.
By root test 
is convergent.
Example: Test the convergence of the series whose nth term is given below-
Sol.
By root test 
is convergent.
Example: show that the following series is convergent.
Sol.
By root test 
is convergent.
Example: Test the convergence of the following series:
Sol. Here, we have,
Therefore the given series is convergent.
Statement- suppose 
is a series of positive terms such that 
then,
1. If k<1 , the series will be convergent.
2. If k>1 , then the series will be divergent.
Proof: Proof:
Case-1:
We know that from the definition of limits ,it follows,
But,
Is the finite quantity. So 
is convergent.
Case-2:
There could be some finite terms in starting which will never satisfy the condition,
In this case we can find a number ‘m’,
Ignoring the first ‘m’ terms, if we write the series as
We have , in this case
So that 
is divergent.
Example-1: Test the convergence of the series whose n’th term is given below-
n’th term = 
Sol. We have 
and 
By D’Alembert ratio test,
So that by D’Alembert ratio test , the series will be convergent.
Example-2: Test for the convergence of the n’th term of the series given below-
Sol. We have ,
Now , by D’Almbert ratio test 
converges if 
and diverges if 
At x = 1 , this test fails.
Now , when x = 1
The limit is finite and not zero.
Then by comparison test, 
converges or diverges together.
Since 
is the form of 
, in which 
Hence 
diverges then 
will also diverge.
Therefore in the given series 
converges if x<1 and diverges if x≥1
The series 
will be
1. Convergent if the value of p is : p>1
2. Divergent if the value of p is: p≤1
Proof;
Let p>1
Similarly,
And so on.
Adding we get,
The right hand side of the above inequality is an infinite geometric series , the sum of the geometric series is finite hence the given series is convergent
Example-1: Test the convergence of the following series:
Sol.
Here we take,
Which not zero and finite ,
So by comparison test , 
and 
both converges or diverges, but by p-series test
Is convergent . So that 
is convergent.
Example-2: Test the convergence of the given series:
Sol. Here,
Take,
is divergent by p-series test. (p = 0 <1)
Example-3: Test the convergence of the series:
Sol.
By p-series test,
Is convergent ( p = 3>1).
Positive term series- if all the terms in an infinite series are positive after few negative terms , then the series said to be a positive term series.
Suppose ,
-22-65+ 55 +69 99+125+………….is a positive term series.
If we remove these negative terms , then the nature of the series does not change.
Comparison test-
Statement-
Suppose we have two series of positive terms 
and 
then,
, where k is a finite number , then both series converges or diverges together.
Proof- we know that by the definition of limits, there exist a positive number epsilon(ε)
Which is very small. Such that
According to definition( comparison test)
|
|< ε for n>m , that means
k-ε < 
for n>m
Ignoring the first m terms of the series,
We get
k-ε < 
for n>m for all n ………………..(1)
There will be two cases-
Case-1: 
is convergent , then
(
) = r (say) , where r is finite number
From (1),
(
)<
(
) = 
Therefore 
is also convergent.
Case-2: : 
is divergent, then
(
)→∞ …………………………..(2)
From eq. (1)
Then
(
)<
(
)
From(2)
(
)→∞
Hence, 
is also divergent.
Example: Test the convergence of the following series.
Sol. We have 
First we will find 
and the 
And
Here, we can see that, the limit is finite and not zero,
Therefore, 
and 
converges or diverges together.
Since 
is of the form 
where p = 2>1
So that , we can say that,
is convergent , so that 
will also be convergent.
Example: Test the convergence of the following series-
Sol. Here we have the series,
Now,
Now comapare 
We can see that the limit is finite and not zero.
Here 
and 
converges or diverges together since ,
is the form of 
here p = 1,
So that,
is divergent then 
is also divergent.
Example: Show that the following series is convergent.
Sol.
Suppose,
Which is finite and not zero.
By comparison test 
and 
converge or diverge together.
But,
Is convergent. So that 
is also convergent.
Example: Test the series:
Sol. The series is,
Now,
Take,
Which is finite and not zero.
Which is finite and not zero.
By comparison test 
and 
converge or diverge together.
But,
Is divergent. ( p = ½)
So that 
is divergent.
First we will understand about alternating series,
When the terms in a series are alternately negative , then the series is known as alternating series.
For example:
Leibnitz test for the convergence of an alternating series:
The given series will be convergent if it follows the rules given below-
1. Each term should be numerically less than its preceeding term 
Example: Test the convergence of the following alternating series:
Sol. Here in the series, we have
First condition-
So that,
|
| > |
|
That means , each term is not numerically less than its preceeding terms.
Now second condition-
Both conditions are not satisfied for convergence.
Hence the given series is not convergent. It is oscillatory.
Example: Test the following series for the convergence-
Sol. We have the given series, now
We see that, this is an alternating series,
Here,
Also,
By Leibinitz’s test the series is convergent.
A series of the form 
Where all the ‘a’ are independent of x, is known as power series in x
Interval for convergence-
In this power series-
D’Almbert’s ratio test-
If 
, then by ratio test, the power series is convergent, when 
is less than 1.
Or in other way if |x|<1/
then the series converges and diverges for other values.
Thus the interval of convergence of power series is- 
Example-1: If the series 
converges, then find the value of x.
Sol. Here 
Then,
By D’Almbert’s ratio test the series is convergent for |x|<1 and divergent if |x|>1.
So at x = 1
The series becomes- 
At x = -1
This is an alternately convergent series.
This is also convergent series, p = 2
Here, the interval of convergence is 
Example-2: If the series 
converges, then find the value of x.
Sol. Here
Then,
By D’Almbert’s ratio test the series is convergent for |
|<1 or |1-x|>1
Or 
At x = 0, the series becomes- 
which is divergent harmonic series.
At x = 2, the series becomes- 
It is an alternate series which is convergent by Leibnitz rule.
So that the series 
.
Taylor’s Theorem-
If f(x + h) is a function of h which can be expanded in the ascending powers of h and is differentiable by any number of times with respect to h, then-
+ …….+ 
+ ……..
Which is called Taylor’s theorem.
If we put x = a, we get-
+ …….+ 
+ …….. (1)
Maclaurin’s Theorem-
If we put a = 0 and h = x then equation(1) becomes-
+ …….
Which is called Maclaurin’s theorem.
Note – if we put h = x - a then there will be the expansion of F(x) in powers of (x – a)
We get-
+ …….
Example-1: Express the polynomial 
in powers of (x-2).
Sol. Here we have,
f(x) = 
Differentiating the function w.r.t.x-
f’(x) = 
f’’(x) = 12x + 14
f’’’(x) = 12
f’’’’(x)=0
Now using Taylor’s theorem-
+ ……. (1)
Here we have, a = 2,
Put x = 2 in the derivatives of f(x), we get-
f(2) = 
f’(2) = 
f’’(2) = 12(2)+14 = 38
f’’’(2) = 12 and f’’’’(2) = 0
Now put a = 2 and substitute the above values in equation(1), we get-
Taylor’s theorem for functions of two variables-
Suppose f(x , y) be a function of two independent variables x and y. Then,
+ ……………
Maclaurin’s series is the special case of Taylor’s series-
When we put a = 0 and b = 0 (about origin) in Taylor’s series, we get-
+ ……………
Example-2: Expand f(x , y) = 
in powers of x and y about origin.
Sol. Here we have the function-
f(x , y) = 
Here , a = 0 and b = 0 then
f(0 , 0) = 
Now we will find partial derivatives of the function-
Now using Taylor’s theorem-
+………
Suppose h = x and k = y , we get
+…….
= 
+……….
Example-3: Find the Taylor’s expansion of 
about (1 , 1) up to second degree term.
Sol. We have,
At (1 , 1)
Now by using Taylor’s theorem-
……
Suppose 1 + h = x then h = x – 1
1 + k = y then k = y - 1
……
= 
……..
Let’s first define the exponential functions-
A function which contains 
where e is the constant, called an exponential function.
Here ‘e’ is the exponent which has an approximate value 2.7183
The power series for 
The value of 
can be defined in terms of the following power series:
If we obtain an actual value of 
by adding all the terms of the series, then the series is said to be convergent.
The more terms we take in the series, the closer will be the value of 
to its actual value.
Example: Find the value of 5
, correct to five decimal places by using the power series for 
.
Sol. As we know that the exponential series is-
Here we get-
Now
Example: Expand 
as far as the term in 
.
Sol. We know that the power series for 
is-
Here we have to find- 
So that-
On solving, we get-
As we know that,
And 
By adding these equations-
And subtracting (2) from (1), we get-
Note- 
Example-1: Verify 
Sol. As we know that 
And 
Hence,
and 
That means
The following function is a logarithmic function-
‘a’ is any value which is greater than 0, except 1
The natural logarithmic function is-
The value of e = 2.71828182….
Properties of logarithms-
1.
2. 
3. 
4. 
then x = y
Example: Simply the logarithm 
Sol. We can write this as-
= 3
Example: Simplify 
Sol. We know that-
So that
