
English: 
Do you know, why 0 factorial is equal to 1
and what is the value of 1 by 2 factorial?
You are watching BRAIN EXPLODERS, the world
where brain starts working. Let us start with
the sum of first ëní natural numbers, S
n equals to 1 plus 2 plus 3 and so on up to
n. So, s 1 is 1. S 2 is the sum of first two
natural numbers that is 1 plus 2 equals to
3. S 10 is the sum of first 10 natural numbers
which is 55. But if someone ask for the value
of s 1.5 or s 2.5 then it means that he is
asking for sum of first 1.5 or 2.5 natural
numbers which is an invalid question. We will
come to it later. Let us first discuss the

English: 
Do you know, why 0 factorial is equal to 1
and what is the value of 1 by 2 factorial?
You are watching BRAIN EXPLODERS, the world
where brain starts working. Let us start with
the sum of first ëní natural numbers, S
n equals to 1 plus 2 plus 3 and so on up to
n. So, s 1 is 1. S 2 is the sum of first two
natural numbers that is 1 plus 2 equals to
3. S 10 is the sum of first 10 natural numbers
which is 55. But if someone ask for the value
of s 1.5 or s 2.5 then it means that he is
asking for sum of first 1.5 or 2.5 natural
numbers which is an invalid question. We will
come to it later. Let us first discuss the

English: 
Do you know, why 0 factorial is equal to 1
and what is the value of 1 by 2 factorial?
You are watching BRAIN EXPLODERS, the world
where brain starts working. Let us start with
the sum of first ëní natural numbers, S
n equals to 1 plus 2 plus 3 and so on up to
n. So, s 1 is 1. S 2 is the sum of first two
natural numbers that is 1 plus 2 equals to
3. S 10 is the sum of first 10 natural numbers
which is 55. But if someone ask for the value
of s 1.5 or s 2.5 then it means that he is
asking for sum of first 1.5 or 2.5 natural
numbers which is an invalid question. We will
come to it later. Let us first discuss the

English: 
Do you know, why 0 factorial is equal to 1
and what is the value of 1 by 2 factorial?
You are watching BRAIN EXPLODERS, the world
where brain starts working. Let us start with
the sum of first ëní natural numbers, S
n equals to 1 plus 2 plus 3 and so on up to
n. So, s 1 is 1. S 2 is the sum of first two
natural numbers that is 1 plus 2 equals to
3. S 10 is the sum of first 10 natural numbers
which is 55. But if someone ask for the value
of s 1.5 or s 2.5 then it means that he is
asking for sum of first 1.5 or 2.5 natural
numbers which is an invalid question. We will
come to it later. Let us first discuss the

English: 
Do you know, why 0 factorial is equal to 1
and what is the value of 1 by 2 factorial?
You are watching BRAIN EXPLODERS, the world
where brain starts working. Let us start with
the sum of first ëní natural numbers, S
n equals to 1 plus 2 plus 3 and so on up to
n. So, s 1 is 1. S 2 is the sum of first two
natural numbers that is 1 plus 2 equals to
3. S 10 is the sum of first 10 natural numbers
which is 55. But if someone ask for the value
of s 1.5 or s 2.5 then it means that he is
asking for sum of first 1.5 or 2.5 natural
numbers which is an invalid question. We will
come to it later. Let us first discuss the

English: 
Do you know, why 0 factorial is equal to 1
and what is the value of 1 by 2 factorial?
You are watching BRAIN EXPLODERS, the world
where brain starts working. Let us start with
the sum of first ëní natural numbers, S
n equals to 1 plus 2 plus 3 and so on up to
n. So, s 1 is 1. S 2 is the sum of first two
natural numbers that is 1 plus 2 equals to
3. S 10 is the sum of first 10 natural numbers
which is 55. But if someone ask for the value
of s 1.5 or s 2.5 then it means that he is
asking for sum of first 1.5 or 2.5 natural
numbers which is an invalid question. We will
come to it later. Let us first discuss the

English: 
general formula for the sum of first n natural
numbers. Write these natural numbers in two
ways. One in ascending and another in descending
order. On adding these equations, we get 2sn
equals to n plus 1 plus n plus 1 plus n plus
1 and so on for n times. Hence, 2 s n is equal
to n times n plus 1 and s n is equal to n
times n plus 1 over 2. This is a function
of n which helps us to find the sum of first
n natural numbers. Since this is a function
of n with a domain of minus infinity to plus
infinity therefore it has a real value for
1.5 also. So if we ignore that n is a natural
number, then we can say that s 1.5 is equal
to 1.875. There is a similar case with the
product of first n natural numbers. There

English: 
general formula for the sum of first n natural
numbers. Write these natural numbers in two
ways. One in ascending and another in descending
order. On adding these equations, we get 2sn
equals to n plus 1 plus n plus 1 plus n plus
1 and so on for n times. Hence, 2 s n is equal
to n times n plus 1 and s n is equal to n
times n plus 1 over 2. This is a function
of n which helps us to find the sum of first
n natural numbers. Since this is a function
of n with a domain of minus infinity to plus
infinity therefore it has a real value for
1.5 also. So if we ignore that n is a natural
number, then we can say that s 1.5 is equal
to 1.875. There is a similar case with the
product of first n natural numbers. There

English: 
general formula for the sum of first n natural
numbers. Write these natural numbers in two
ways. One in ascending and another in descending
order. On adding these equations, we get 2sn
equals to n plus 1 plus n plus 1 plus n plus
1 and so on for n times. Hence, 2 s n is equal
to n times n plus 1 and s n is equal to n
times n plus 1 over 2. This is a function
of n which helps us to find the sum of first
n natural numbers. Since this is a function
of n with a domain of minus infinity to plus
infinity therefore it has a real value for
1.5 also. So if we ignore that n is a natural
number, then we can say that s 1.5 is equal
to 1.875. There is a similar case with the
product of first n natural numbers. There

English: 
general formula for the sum of first n natural
numbers. Write these natural numbers in two
ways. One in ascending and another in descending
order. On adding these equations, we get 2sn
equals to n plus 1 plus n plus 1 plus n plus
1 and so on for n times. Hence, 2 s n is equal
to n times n plus 1 and s n is equal to n
times n plus 1 over 2. This is a function
of n which helps us to find the sum of first
n natural numbers. Since this is a function
of n with a domain of minus infinity to plus
infinity therefore it has a real value for
1.5 also. So if we ignore that n is a natural
number, then we can say that s 1.5 is equal
to 1.875. There is a similar case with the
product of first n natural numbers. There

English: 
general formula for the sum of first n natural
numbers. Write these natural numbers in two
ways. One in ascending and another in descending
order. On adding these equations, we get 2sn
equals to n plus 1 plus n plus 1 plus n plus
1 and so on for n times. Hence, 2 s n is equal
to n times n plus 1 and s n is equal to n
times n plus 1 over 2. This is a function
of n which helps us to find the sum of first
n natural numbers. Since this is a function
of n with a domain of minus infinity to plus
infinity therefore it has a real value for
1.5 also. So if we ignore that n is a natural
number, then we can say that s 1.5 is equal
to 1.875. There is a similar case with the
product of first n natural numbers. There

English: 
general formula for the sum of first n natural
numbers. Write these natural numbers in two
ways. One in ascending and another in descending
order. On adding these equations, we get 2sn
equals to n plus 1 plus n plus 1 plus n plus
1 and so on for n times. Hence, 2 s n is equal
to n times n plus 1 and s n is equal to n
times n plus 1 over 2. This is a function
of n which helps us to find the sum of first
n natural numbers. Since this is a function
of n with a domain of minus infinity to plus
infinity therefore it has a real value for
1.5 also. So if we ignore that n is a natural
number, then we can say that s 1.5 is equal
to 1.875. There is a similar case with the
product of first n natural numbers. There

English: 
exists a function which can help us in finding
this product. This function is Gamma function.
Gamma n equals to integration 0 to infinity
e to the power minus x times x to the power
n minus 1 dx. On applying integration by parts
assuming that x to the power n minus 1 is
first function and e to the minus x is second
function, we get gamma n equals to n minus
1 times gamma n minus 1. With the help of
this property we can write n equals to n minus
1 times n minus 2 times n minus 3 and so on
up to 2 times 1 times gamma 1. Value of gamma
n can be easily calculated as 1. We get gamma
n equals to factorial n minus 1. On putting
n equals to 1 we see that factorial 0 equals
to 1. To can calculate the value of 1 by 2
factorial we can put n equals to 3 by 2 but
this function becomes a little bit complicated

English: 
exists a function which can help us in finding
this product. This function is Gamma function.
Gamma n equals to integration 0 to infinity
e to the power minus x times x to the power
n minus 1 dx. On applying integration by parts
assuming that x to the power n minus 1 is
first function and e to the minus x is second
function, we get gamma n equals to n minus
1 times gamma n minus 1. With the help of
this property we can write n equals to n minus
1 times n minus 2 times n minus 3 and so on
up to 2 times 1 times gamma 1. Value of gamma
n can be easily calculated as 1. We get gamma
n equals to factorial n minus 1. On putting
n equals to 1 we see that factorial 0 equals
to 1. To can calculate the value of 1 by 2
factorial we can put n equals to 3 by 2 but
this function becomes a little bit complicated

English: 
exists a function which can help us in finding
this product. This function is Gamma function.
Gamma n equals to integration 0 to infinity
e to the power minus x times x to the power
n minus 1 dx. On applying integration by parts
assuming that x to the power n minus 1 is
first function and e to the minus x is second
function, we get gamma n equals to n minus
1 times gamma n minus 1. With the help of
this property we can write n equals to n minus
1 times n minus 2 times n minus 3 and so on
up to 2 times 1 times gamma 1. Value of gamma
n can be easily calculated as 1. We get gamma
n equals to factorial n minus 1. On putting
n equals to 1 we see that factorial 0 equals
to 1. To can calculate the value of 1 by 2
factorial we can put n equals to 3 by 2 but
this function becomes a little bit complicated

English: 
exists a function which can help us in finding
this product. This function is Gamma function.
Gamma n equals to integration 0 to infinity
e to the power minus x times x to the power
n minus 1 dx. On applying integration by parts
assuming that x to the power n minus 1 is
first function and e to the minus x is second
function, we get gamma n equals to n minus
1 times gamma n minus 1. With the help of
this property we can write n equals to n minus
1 times n minus 2 times n minus 3 and so on
up to 2 times 1 times gamma 1. Value of gamma
n can be easily calculated as 1. We get gamma
n equals to factorial n minus 1. On putting
n equals to 1 we see that factorial 0 equals
to 1. To can calculate the value of 1 by 2
factorial we can put n equals to 3 by 2 but
this function becomes a little bit complicated

English: 
exists a function which can help us in finding
this product. This function is Gamma function.
Gamma n equals to integration 0 to infinity
e to the power minus x times x to the power
n minus 1 dx. On applying integration by parts
assuming that x to the power n minus 1 is
first function and e to the minus x is second
function, we get gamma n equals to n minus
1 times gamma n minus 1. With the help of
this property we can write n equals to n minus
1 times n minus 2 times n minus 3 and so on
up to 2 times 1 times gamma 1. Value of gamma
n can be easily calculated as 1. We get gamma
n equals to factorial n minus 1. On putting
n equals to 1 we see that factorial 0 equals
to 1. To can calculate the value of 1 by 2
factorial we can put n equals to 3 by 2 but
this function becomes a little bit complicated

English: 
exists a function which can help us in finding
this product. This function is Gamma function.
Gamma n equals to integration 0 to infinity
e to the power minus x times x to the power
n minus 1 dx. On applying integration by parts
assuming that x to the power n minus 1 is
first function and e to the minus x is second
function, we get gamma n equals to n minus
1 times gamma n minus 1. With the help of
this property we can write n equals to n minus
1 times n minus 2 times n minus 3 and so on
up to 2 times 1 times gamma 1. Value of gamma
n can be easily calculated as 1. We get gamma
n equals to factorial n minus 1. On putting
n equals to 1 we see that factorial 0 equals
to 1. To can calculate the value of 1 by 2
factorial we can put n equals to 3 by 2 but
this function becomes a little bit complicated

English: 
to calculate. To solve it, we have to convert
it to some simple forms. Put x equals to t
square and dx equals to 2tdt. Since, using
any other variable doesnít affect the value
of integral therefore we can use x and y instead
of t. Multiply equation 1 and 2. Since, x
and y are independent of each other therefore
we can write it as double integral. Donít
worry if you are not familiar with double
integrals because we are not going to use
any complicated property of it. Just convert
it to polar form. For this we put r cos theta,
r sin theta and r dr d theta in place of x,
y and dxdy respectively. X and y lie between

English: 
to calculate. To solve it, we have to convert
it to some simple forms. Put x equals to t
square and dx equals to 2tdt. Since, using
any other variable doesnít affect the value
of integral therefore we can use x and y instead
of t. Multiply equation 1 and 2. Since, x
and y are independent of each other therefore
we can write it as double integral. Donít
worry if you are not familiar with double
integrals because we are not going to use
any complicated property of it. Just convert
it to polar form. For this we put r cos theta,
r sin theta and r dr d theta in place of x,
y and dxdy respectively. X and y lie between

English: 
to calculate. To solve it, we have to convert
it to some simple forms. Put x equals to t
square and dx equals to 2tdt. Since, using
any other variable doesnít affect the value
of integral therefore we can use x and y instead
of t. Multiply equation 1 and 2. Since, x
and y are independent of each other therefore
we can write it as double integral. Donít
worry if you are not familiar with double
integrals because we are not going to use
any complicated property of it. Just convert
it to polar form. For this we put r cos theta,
r sin theta and r dr d theta in place of x,
y and dxdy respectively. X and y lie between

English: 
to calculate. To solve it, we have to convert
it to some simple forms. Put x equals to t
square and dx equals to 2tdt. Since, using
any other variable doesnít affect the value
of integral therefore we can use x and y instead
of t. Multiply equation 1 and 2. Since, x
and y are independent of each other therefore
we can write it as double integral. Donít
worry if you are not familiar with double
integrals because we are not going to use
any complicated property of it. Just convert
it to polar form. For this we put r cos theta,
r sin theta and r dr d theta in place of x,
y and dxdy respectively. X and y lie between

English: 
to calculate. To solve it, we have to convert
it to some simple forms. Put x equals to t
square and dx equals to 2tdt. Since, using
any other variable doesnít affect the value
of integral therefore we can use x and y instead
of t. Multiply equation 1 and 2. Since, x
and y are independent of each other therefore
we can write it as double integral. Donít
worry if you are not familiar with double
integrals because we are not going to use
any complicated property of it. Just convert
it to polar form. For this we put r cos theta,
r sin theta and r dr d theta in place of x,
y and dxdy respectively. X and y lie between

English: 
to calculate. To solve it, we have to convert
it to some simple forms. Put x equals to t
square and dx equals to 2tdt. Since, using
any other variable doesnít affect the value
of integral therefore we can use x and y instead
of t. Multiply equation 1 and 2. Since, x
and y are independent of each other therefore
we can write it as double integral. Donít
worry if you are not familiar with double
integrals because we are not going to use
any complicated property of it. Just convert
it to polar form. For this we put r cos theta,
r sin theta and r dr d theta in place of x,
y and dxdy respectively. X and y lie between

English: 
0 to infinity hence any point can lie anywhere
in first quadrant. For first quadrant, theta
lies between 0 to pi by 2 and r lies between
0 to infinity so limits of integrals are changed
accordingly. Substitute r square equals to
t and r dr equals to dt by 2. Since t and
theta are independent of each other therefore
we can calculate the values of these integrals
separately and finally we get factorial 1
by 2 equals to root pi by 2. Gamma function
can be written in different forms. Some of
these are given on the screen. Donít forget
to like and share our video and for more such
videos subscribe to our channel BRAIN EXPLODERS.
Thanks for watching.

English: 
0 to infinity hence any point can lie anywhere
in first quadrant. For first quadrant, theta
lies between 0 to pi by 2 and r lies between
0 to infinity so limits of integrals are changed
accordingly. Substitute r square equals to
t and r dr equals to dt by 2. Since t and
theta are independent of each other therefore
we can calculate the values of these integrals
separately and finally we get factorial 1
by 2 equals to root pi by 2. Gamma function
can be written in different forms. Some of
these are given on the screen. Donít forget
to like and share our video and for more such
videos subscribe to our channel BRAIN EXPLODERS.
Thanks for watching.

English: 
0 to infinity hence any point can lie anywhere
in first quadrant. For first quadrant, theta
lies between 0 to pi by 2 and r lies between
0 to infinity so limits of integrals are changed
accordingly. Substitute r square equals to
t and r dr equals to dt by 2. Since t and
theta are independent of each other therefore
we can calculate the values of these integrals
separately and finally we get factorial 1
by 2 equals to root pi by 2. Gamma function
can be written in different forms. Some of
these are given on the screen. Donít forget
to like and share our video and for more such
videos subscribe to our channel BRAIN EXPLODERS.
Thanks for watching.

English: 
0 to infinity hence any point can lie anywhere
in first quadrant. For first quadrant, theta
lies between 0 to pi by 2 and r lies between
0 to infinity so limits of integrals are changed
accordingly. Substitute r square equals to
t and r dr equals to dt by 2. Since t and
theta are independent of each other therefore
we can calculate the values of these integrals
separately and finally we get factorial 1
by 2 equals to root pi by 2. Gamma function
can be written in different forms. Some of
these are given on the screen. Donít forget
to like and share our video and for more such
videos subscribe to our channel BRAIN EXPLODERS.
Thanks for watching.

English: 
0 to infinity hence any point can lie anywhere
in first quadrant. For first quadrant, theta
lies between 0 to pi by 2 and r lies between
0 to infinity so limits of integrals are changed
accordingly. Substitute r square equals to
t and r dr equals to dt by 2. Since t and
theta are independent of each other therefore
we can calculate the values of these integrals
separately and finally we get factorial 1
by 2 equals to root pi by 2. Gamma function
can be written in different forms. Some of
these are given on the screen. Donít forget
to like and share our video and for more such
videos subscribe to our channel BRAIN EXPLODERS.
Thanks for watching.

English: 
0 to infinity hence any point can lie anywhere
in first quadrant. For first quadrant, theta
lies between 0 to pi by 2 and r lies between
0 to infinity so limits of integrals are changed
accordingly. Substitute r square equals to
t and r dr equals to dt by 2. Since t and
theta are independent of each other therefore
we can calculate the values of these integrals
separately and finally we get factorial 1
by 2 equals to root pi by 2. Gamma function
can be written in different forms. Some of
these are given on the screen. Donít forget
to like and share our video and for more such
videos subscribe to our channel BRAIN EXPLODERS.
Thanks for watching.
