What is the sum of these infinite series in statistics?

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SUMMARY

The discussion centers on calculating the sums of specific infinite series in statistics, particularly focusing on E(1/n^2) from 1 to infinity, E(1/(L+n^2)) from 0 to infinity, and E(a^n) from 0 to infinity where a is between 0 and 1. The first series, E(1/n^2), converges to a known value of π²/6. The second series, E(1/(L+n^2)), lacks a simple closed form, while the third series converges to 1/(1-a) for 0 < a < 1, which is a crucial result for statistical applications.

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freedominator
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My statistics professor wanted me to remember an infinite series sum, but i can't remember what it was
i think it was either ( E(1/n^2) 1 to infinite ) or ( E(1/(L+n^2)) 0 to infinite ) or ( E(a^n) 0 to infinite where n is between 0 and 1)
i think it was the 2nd one, does anyone know how to calculate either of these?
 
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E should be the sum symbol(##\Sigma##)?
The first one has a simple solution, I don't think I ever saw any closed form for the second.
At the third one, do you mean "a is between 0 and 1"? In that case, it has a nice value as well, which is very useful to know.
 

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