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Intro — Math with Python: Pi (π) and Pascal’s Triangle
- The article proposes a new formula to approximate the value of π, inspired by Nilakantha’s Series.
- The formula is simpler and more elegant than the series method and is derived from Pascal's Triangle.
- The connection of π with Pascal's Triangle is established where a 'Reverse of the Sum' pattern is discovered.
- Odd-numbered rows of Pascal's Triangle can be used to estimate the value of π and reach an accuracy of 15 decimal places.
- The article uses Python code to verify the new formula and the connection between π and Pascal's Triangle.
- A formal proof is provided for the 'Reverse of the Sum' property of the Pascal's Triangle.
- The new formula and Nilakantha’s series are found to be mathematically equivalent, with the same convergence rate.
- A table and chart are shared to show the convergence rate and the number of terms needed to achieve a specified level of precision.
- The skipped rows in Pascal's Triangle correspond to the 'base' term of 3 in the formula, and to reach a precision of n decimal places, 3 + 2n rows are required.
- The article concludes that the new formula and the connection with Pascal's Triangle provide a novel perspective on the value of π.
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