Concerning Pi, again

For Pi Day this year (2018), I will provide some results related to this interesting mathematical constant. These results are mostly inspired or taken from answers I gave at quora.com about  \pi or about similar mathematical topics.

The millionth decimal digit of π is 1 (verified with Mathematica).

The 10 millionth decimal digit of π is found to be 7, and the 100 millionth decimal digit of π is 2.

The billionth decimal digit of \pi (in base 10) is 9 (verified with Mathematica).

The 2 billionth decimal digit of π is found to be 0 (this result takes a longer time to compute with Mathematica).

Here are some (repeated) number sequences or numeric strings found among the first 2 billion decimal digits of π.

The numeric string 777777777 appears at the 24, 658, 601 st decimal digit of π :

9919408245759718530477777777724846769425931046864

The numeric string 111111111 appears at the 812, 432, 526 th decimal digit of π :

2891450444990691713511111111100399876875718824885

Here are also two numeric strings from 1 to 9 in increasing order and decreasing order:

The numeric string 123456789 appears at the 523, 551, 502 nd decimal digit of π:

7260489917323889207212345678922486448188070486710

The numeric string 987654321 appears at the 719, 473, 323 rd decimal digit of π :

5221398663241526793698765432160793011913242320510

The numeric strings above can be calculated or found with the help of the following link or web page:

Irrational Numbers Search Engine

The numerical value of \pi^{\pi} to 1000 decimal digits is equal to:

36.46215960720791177099082602269212366636550840222881873870933592293407436888169990462007987570677485436814688343670070542736699139359264431565675267180230917777595737242260530320050233549595161382594571885422222305402433199779769167302876444780028452117394296018175249159350019492001619423210110480018557258718860782819839215304503453543238476218257664861595609057280314341958390400811991506636066295817900302292747422204210046403709493285441101884797707466358510710362803891181156618083260884536505255311095948029552909133361385823497120761861157606574436205295895657736468959837126404885207348833917602169536002174958035720670509318770633086434935593202425189496008880555048213388792769304015639745480898380866639428337794250284522113741878602793251848366623602214652151453276098545038540482041645516919082097210265379423765817354600472953993840487842116366153365057093066926223775915023204727672609958737278566633593210689698807508602353552267490321670730929373240451651980750157959708689483469068

Two expressions involving π and infinite sums:

pi infinite cums

Representation of π in continued fraction form:

pi continued frac form

The sum of π and e, the base of natural logarithms, is equal to:

\pi + e=\displaystyle  \sum _{k=0}^{\infty } \frac{(3 k)^2+1}{(3 k)!}+2 i \ln \left(\frac{1-i}{1+i}\right)i

The letter i  represents the imaginary unit of complex numbers.

Another expression involving π, e, and an infinite product:

\pi = \displaystyle 2 e \prod _{k=1}^{\infty } \left(\frac{2}{k}+1\right)^{(-1)^{k+1} k}

And here is an identity relating the Golden Ratio, π, e, and the imaginary unit i:

\displaystyle \varphi=e^{i\pi/5}+e^{-i\pi/5}=\frac{1+\sqrt{5}}{2}

The value of π can be deduced from the identity above:

\displaystyle\pi =5 i \ln \left(\frac{1}{2} \left(\varphi -\sqrt{\varphi ^2-4}\right)\right)

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