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Infinite Series
ఊపిరితిత్తులకు కూడా అంతే కాక ఆంధ్ర oi oo ok o oh o ఊపు
Example 1: Ramanujan's Formula for
One of Ramanujan's most famous infinite series is a rapidly converging formula for , which is used in modern computations of :
\frac{1}{\pi} = \frac{2\sqrt{2}}{9801} \sum_{k=0}^\infty \frac{(4k)!(1103 + 26390k)}{(k!)^4 396^{4k}}
Verification for :
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\pi \approx \frac{9801}{2\sqrt{2} \cdot 1103} \approx 3.141592730
This is accurate to 6 decimal places with only one term!
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Example 2: Series for (Euler's Number)
Ramanujan developed unique series for , including:
e = \sum_{n=0}^\infty \frac{1}{n!}
Verify for terms:
e \approx 1 + 1 + \frac{1}{2} + \frac{1}{6} + \frac{1}{24} + \frac{1}{120} = 2.7166667
This approximation is close to the true value .
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Continued Fractions
Continued fractions are expressions of numbers as an infinite sequence of nested fractions. Ramanujan contributed extensively to this field by deriving elegant representations of numbers and functions.
Example 1: Ramanujan's Continued Fraction for
For , Ramanujan proposed:
e^{\pi} = 2 + \cfrac{1}{3 + \cfrac{1}{7 + \cfrac{1}{15 + \cfrac{1}{1 + \dots}}}}
This representation reveals the deep spltructure behind exponential functions and their relationships.
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Example 2: Golden Ratio
Ramanujan explored continued fractions for the golden ratio , where:
\phi = 1 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{1 + \dots}}}
This infinite continued fraction converges to , a value crucial in art, architecture, and nature.
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Applications
1. Infinite Series:
Used for high-precision calculations in numerical analysis.
Ramanujan's series for is the basis for many modern algorithms.
2. Continued Fractions:
Widely used in numerical approximations, cryptography, and computer science.
Ramanujan's fractions have applications in approximating irrational numbers and studying special functions.
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Significance
Ramanujan’s work on infinite series and continued fractions demonstrated his unparalleled ability to find hidden patterns in numbers. These discoveries bridge pure mathematics and applied fields, offering elegant solutions to complex problems. His methods and insights continue to inspire mathematicians and scientists worldwide.