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I was trying various methods to implement a program that gives the digits of pi sequentially. I tried the Taylor series method, but it proved to converge extremely slowly (when I compared my result with the online values after some time). Anyway, I am trying better algorithms.

So, while writing the program I got stuck on a problem, as with all algorithms: How do I know that the n digits that I've calculated are accurate?

Undoubtedly, for your purposes (which I assume is just a programming exercise), the best thing is to check your results against any of the listings of the digits of pi on the web.

And how do we know that those values are correct? Well, I could say that there are computer-science-y ways to prove that an implementation of an algorithm is correct.

More pragmatically, if different people use different algorithms, and they all agree to (pick a number) a thousand (million, whatever) decimal places, that should give you a warm fuzzy feeling that they got it right.

Historically, William Shanks published pi to 707 decimal places in 1873. Poor guy, he made a mistake starting at the 528th decimal place.

Very interestingly, in 1995 an algorithm was published that had the property that would directly calculate the nth digit (base 16) of pi without having to calculate all the previous digits!

Finally, I hope your initial algorithm wasn't pi/4 = 1 - 1/3 + 1/5 - 1/7 + ... That may be the simplest to program, but it's also one of the slowest ways to do so. Check out the pi article on Wikipedia for faster approaches.

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