Explain the formula for calculating π presented by Ramanujan.
Understand the Problem
The image contains a mathematical formula related to the calculation of π (pi) attributed to Srinivasa Ramanujan. It discusses an infinite series that converges to π and includes factorials and constants. The question likely regards the understanding or implications of this formula.
Answer
The value of $\pi$ is given by the formula: $$ \pi = \frac{9801}{1103\sqrt{8}} $$
Answer for screen readers
The value of $\pi$ is given by the formula: $$ \pi = \frac{9801}{1103\sqrt{8}} $$
Steps to Solve
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Understanding the series The series given is an infinite series to calculate $\pi$. It has a specific form involving factorials and powers of numbers with constants.
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Identifying the series components The main components of the series are:
- $(4n)!$ is the factorial of $4n$.
- The term $(1103 + 26390n)$ is linear in $n$.
- The denominator includes the $(n!)^4$ and the term $396^{4n}$.
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Writing the series explicitly The series can be expressed as: $$ S = \sum_{n=0}^{\infty} \frac{(4n)!(1103 + 26390n)}{(n!)^4 (396)^{4n}} $$
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Multiplying by the factor The series must be multiplied by the prefactor $\frac{2\sqrt{2}}{9801}$: $$ A = \frac{2\sqrt{2}}{9801} S $$
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Setting the formula for $\pi$ The formula then simplifies and sets: $$ \frac{A}{\pi} = \frac{9801}{1103\sqrt{8}} $$
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Calculating the value of $\pi$ To find $\pi$, we take the reciprocal of the right side: $$ \pi = \frac{9801}{1103\sqrt{8}} $$
The value of $\pi$ is given by the formula: $$ \pi = \frac{9801}{1103\sqrt{8}} $$
More Information
This formula comes from a series derived by Srinivasa Ramanujan, who contributed immensely to the understanding of mathematical analysis and number theory. This particular series converges very rapidly to $\pi$, which showcases Ramanujan's deep insight into the nature of mathematical constants.
Tips
- Failing to properly evaluate factorials when $n$ becomes large can lead to errors in calculating the series.
- It's important to correctly apply the series formula and ensure that the entire series is considered when calculating $\pi$.
- Missing the multiplication factor $\frac{2\sqrt{2}}{9801}$ can lead to incorrect conclusions about the relationship between the series and $\pi$.
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