Quickly Estimate Apery's Constant Using Partial Sums and Integrals

In summary, the conversation discusses estimating the value of the series 1 + \frac{1}{2^3} + \frac{1}{3^3} + \frac{1}{4^3} + \frac{1}{5^3} + ... in 1 minute, using the concept of the Riemann zeta constant. One person suggests using an integral to estimate the remainder of the series, with m=2 yielding a sum of about 1.18 and m=3 yielding a sum of 1.19. The other person hopes for a more interesting approach.
  • #1
snipez90
1,101
5
Estimate [itex]1 + \frac{1}{2^3} + \frac{1}{3^3} + \frac{1}{4^3} + \frac{1}{5^3} + ... [/itex] in 1 minute (See "[URL [Broken] constant[/URL]).

I couldn't think of a clever way to quickly do this. Any ideas?
 
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  • #2
after thinking about this for more than one minute :rolleyes: I am convinced that the best I could hope to do was sum a few terms and estimate the remainder using an integral. Something like
[tex]
Ʃ_{n=1}^{\infty} n^{-3} \approx Ʃ_{n=1}^{m} n^{-3} + ∫_{m+1}^{\infty} x^{-3} dx = Ʃ_{n=1}^{m} n^{-3} + (m+1)^{-2}/2.[/tex]

m=2 yields the sum is about 1.18, m=3 yields 1.19, so I would guess 1.2 for at most two significant figures. Hopefully someone else has something more interesting than that ...
 

1. What is Apery's Constant?

Apery's Constant, denoted by ζ(3), is a mathematical constant that is approximately equal to 1.2020569. It is also known as Apéry's Number or the Apery's Number.

2. Who discovered Apery's Constant?

Apery's Constant was discovered by the French mathematician Roger Apéry in 1978.

3. Why is Apery's Constant important?

Apery's Constant is important because it is a fundamental constant in mathematics, appearing in many different fields such as number theory, calculus, and physics. It is also closely related to other important constants such as π and e.

4. How is Apery's Constant estimated?

The most commonly used method for estimating Apery's Constant is by using the Euler-Maclaurin formula and calculating the sum of the first n terms of the series ζ(3). As n approaches infinity, the estimate gets closer to the true value of Apery's Constant.

5. What is the current estimate of Apery's Constant?

The current estimate of Apery's Constant is approximately 1.2020569, which has been computed using advanced mathematical techniques and high-precision calculations.

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