Find the 2013th digit after the decimal point in the expression of...

Let $\displaystyle m=\frac{\sqrt{10^{4024}-1}}{3}$

Find the $\displaystyle 2013th$ digit after the decimal point in the expression of \sqrt{m}

It seems a similar problem.

xkcd • View topic - math problem - driving me crazy

But I don't think so. I can't write m to $\displaystyle (\sqrt{a}+\sqrt{b})^n$ a,b,n is integer numbers

Re: Find the 2013th digit after the decimal point in the expression of...

Quote:

Originally Posted by

**coriander** Let $\displaystyle m=\frac{\sqrt{10^{4024}-1}}{3}$

Find the $\displaystyle 2013th$ digit after the decimal point in the expression of \sqrt{m}

It seems a similar problem.

xkcd • View topic - math problem - driving me crazy But I don't think so. I can't write m to $\displaystyle (\sqrt{a}+\sqrt{b})^n$ a,b,n is integer numbers

Do you know the **binomial series**?

Write $\displaystyle \frac{\sqrt{10^{4024}-1}}{3}=\tfrac{1}{3}\left(\sqrt{10^{4024}-1\right)^{\frac{1}{2}}$

**BTW**

When writing an exponent with ore than one character in LaTeX use {} on the exponent.

[tex]10^{4024}[/tex] gives $\displaystyle 10^{4024} $ .

Re: Find the 2013th digit after the decimal point in the expression of...

Can you show me more detail?

Re: Find the 2013th digit after the decimal point in the expression of...

Quote:

Originally Posted by

**coriander** Can you show me more detail?

No, I am not going to do that. It is too darn messy.

You can look up binomial series.

Re: Find the 2013th digit after the decimal point in the expression of...

You might want to specifically look up the **generalised** binomial series since this has a fractional exponent. It's about 2/3 of the way down this page:

Binomial theorem - Wikipedia, the free encyclopedia