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

• Jan 18th 2013, 03:19 AM
coriander
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 &bull; 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
• Jan 18th 2013, 06:28 AM
Plato
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.
$$10^{4024}$$ gives $\displaystyle 10^{4024}$ .
• Jan 18th 2013, 07:03 AM
coriander
Re: Find the 2013th digit after the decimal point in the expression of...
Can you show me more detail?
• Jan 18th 2013, 07:21 AM
Plato
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.
• Jan 18th 2013, 07:48 AM
HallsofIvy
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