Stirling approximation

Says that

ln(n!) ~ n(ln(n)-1)+\frac{1}{2}ln(2πn)

As n goes to infinity

# Thread to share Good Questions from Calculus which requires deep thinking

**Satyendra_2019**#369

**Tushar_2019**#373

Yup, its correct but i am curious in knowing the method @Rohan_Shinde1(if u know other than Leibnitz's)

**arush_2019**#374

@Rohan_Shinde1

For #335

I m getting

f(x)=(1-i)× \left( { \frac {(1+i)^n + (1-i)^n}{2} } \right)

Don't know how to proceed

**Rohan_Shinde1**#375

Differentiate both sides wrt x and don't think that a differential equation will be formed. The question seems to be very well framed that no DE forms and we get the function directly.

@arush_kumar_singh

But that would turn the sum as a complex number which it isn't.

**arush_2019**#376

@Tushar_Rathore

It's simple jee ques take e^x out from integral

And diffrentiate...

**arush_2019**#378

From here we get that

\phi_0=1 , \phi_1=1 , \phi_2 =-1 , \phi_3 = -1 , \phi_4=+1 , \phi_5 = +1 ....

So we have to calculate

\color{red} '^nC_0+'^nC_1-'^nC_2 - '^nC_3 +'^nC_4 +'^nC_5 - '^nC_6 - '^nC_7....

Am I doing correct???

@Rohan_Shinde1

**Rohan_Shinde1**#380

Yep got the same \left(\frac 13 \ln 6\right). @Yash_Srivastava2 @Tushar_Rathore

**Rohan_Shinde1**#383

Answer is \displaystyle \frac {1}{2018} B\left(\frac {1}{2018},\frac {1}{2017}+1\right)- \frac {1}{2017} B\left(\frac {1}{2017},\frac {1}{2018}+1\right)

**Tushar_2019**#385

Its in our jee course @Rohan_Shinde1 bro , so answer should also be related to jee

Options are

(1) 2018

(2)2017

(3)0

(4)2019

**arush_2019**#387

@Rohan_Shinde1

Is beta function converted to gamma function..

I think for \int _0^{ \frac π2 } sin^nx × cos^mx we use beta function...

But I don't know how to use it in algebraic quantity

@Tushar_Rathore

I think there should be more options like 1 ; -1

**Rohan_Shinde1**#388

@Tushar_Rathore 0

@arush_kumar_singh

The beta function and Gamma function do not have limited use bro. In higher studies you will find that Gamma function is used in many much topics like Hyperfactorial, G-Barne's Function , K -Function , Zeta function, Eta function, Gaussian integrals and many other similar integrals. It is also used in some very important theorems and transforms like Ramanujan's Master theorem, Laplace transforms , Mellin transform, Fourier transform, etc. It also has wide applications in physics too