Help Calculating Pi using Arctangent formula

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SUMMARY

The discussion focuses on calculating the value of pi using the Arctangent formula: pi = 16 * arctan(1/5) - 4 * arctan(1/239). A user attempted this calculation using MAPLE 13 but received an incorrect result of 3.17 instead of the expected 3.14. Another participant suggested using Mathematica for accurate results and recommended enclosing the arguments of the evalf function in brackets. Additionally, they advised using the remainder theorem for Taylor expansions to determine the necessary number of terms for achieving 53 significant digits.

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  • Understanding of the Arctangent formula and its application in calculating pi.
  • Familiarity with Taylor series and the remainder theorem.
  • Experience with MAPLE 13 and Mathematica software.
  • Knowledge of numerical precision and significant digits in calculations.
NEXT STEPS
  • Learn how to implement the Arctangent formula in MAPLE 13 correctly.
  • Explore the use of Mathematica for numerical computations involving pi.
  • Study the remainder theorem for Taylor expansions to optimize series calculations.
  • Investigate techniques for achieving high precision in numerical analysis.
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Mathematicians, computer scientists, and students working on numerical methods for calculating pi, particularly those using MAPLE or Mathematica for their computations.

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Homework Statement


Using the Arctangent formula
pi = 16 * arctan (1 / 5) - 4*arctan(1 / 239) to calculate the value of pi to 53 significant digits.

Homework Equations



The power series of arctangent(x) is = x − x^3/3 + x^5/5 − x^7/7 + x^9/9...

The Attempt at a Solution



http://tinypic.com/r/fy1tax/5

fy1tax.jpg


I attempted this question on MAPLE 13(shown above), but I get an obvious incorrect answer 3.17... instead of 3.14...

Am i using the formula correctly? I've double checked the values and don't think I missed an exponent or inserted the wrong sign.

-Thanks in advance.
 
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I'm not familiar with Maple, so I can't tell you what it's doing. However, using the same methid in Mathematica, I get 3.14... as expected, so I assume there is something odd about the way you've told Maple to calculate it...perhaps enclosing everything after "evalf" in a pair of brackets might change things?

In any case, I'm not sure you need terms all the way up to x^{51} to get an accuracy of 53 decimal places. Use the remainder theorem for Taylor expansions to figure out how many terms you actually need.
 

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