Quickly Estimate Apery's Constant Using Partial Sums and Integrals

[snipez90]

Estimate $1 + \frac{1}{2^3} + \frac{1}{3^3} + \frac{1}{4^3} + \frac{1}{5^3} + ...$ in 1 minute (See "[URL constant[/URL]).

I couldn't think of a clever way to quickly do this. Any ideas?

 

[jasonRF]

after thinking about this for more than one minute 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
$$Ʃ_{n=1}^{\infty} n^{-3} \approx Ʃ_{n=1}^{m} n^{-3} + ∫_{m+1}^{\infty} x^{-3} dx = Ʃ_{n=1}^{m} n^{-3} + (m+1)^{-2}/2.$$

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 ...