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# Remainder term watch

1. A friend asked me this question, and neither of us could see where he went wrong (didn't help that I don't know much about remainder terms in Taylor expansions), so any input from anyone here would be appreciated.

The expansion

can be used to approximate by substituting in x=1, to give

The remainder term of this expansion is bounded by

The question is to work out how many terms you need to get pi accurate to 10 decimal places.
So we set

(as 1^2n+1 = 1, and the expansion has been multiplied through by 4), to get n is approximately , which is how many terms you'd need to get pi accurate enough.
Can anyone see where we went wrong?
2. My guess it that to be accurate to 10 decimal places, you check to the 11th place, so change that 10^(-10) to 10^(-11) and then if you ignore the 4 you get the answer. I can't think why you ignore the 4 though.
3. Ooh, hadn't thought of that.
Could you explain ignoring the 4 by saying that you need 5*10^10 terms to approximate pi/4; and then you can multiply what you get through by 4 as it's an exact constant and not part of the approximation?
4. That's what I thought, yeah. It's kind of the same as saying if you have 1/12 accurate to 10 places then you can multiply by 4 to get 1/3 to 10 places. Except when I do that on a calculator I only get 1/3 accurate to 9 places.
5. Okay, cheers

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