- #1
obelu
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Please I have a similar problem, how can I compute the sum of this infinite series:
SUM(X / (Y^X) ); where X(i)>0
SUM(X / (Y^X) ); where X(i)>0
The formula for the sum of infinite series 1/n^2 is ∑(1/n^2) = π^2/6. This is known as the Basel problem and was famously solved by Leonhard Euler in 1734.
An infinite series is a sum of an infinite number of terms, with each term being added to the previous one. It can be represented as ∑(a_n), where n represents the number of terms and a_n represents the value of each term.
Euler's proof for the Basel problem involves using the Euler product formula and the gamma function. He showed that ∑(1/n^2) can be expressed as ∏(1 - 1/p^2), where p represents all prime numbers. This can then be simplified to ∏(1 - 1/p^2) = (1 - 1/2^2)(1 - 1/3^2)(1 - 1/5^2)... = π^2/6.
The sum of infinite series 1/n^2 is convergent. This means that the sum of all the terms approaches a finite value as the number of terms increases. In this case, the sum approaches π^2/6 as the number of terms goes to infinity.
The sum of infinite series 1/n^2 has several applications in mathematics, including in the study of harmonic functions and Fourier series. It is also used in physics to calculate the Casimir force, which is a quantum mechanical effect between two parallel plates. Additionally, the value of π^2/6 is used in various calculations and equations in engineering and science.