How does WolframAlpha or other software get such precise values for zeta(3), etc.?

Question

I've been looking with a friend at the values of zeta at the odd integers. WolframAlpha can give us over 100 digits in a second or two, but it seems that if you take the sum out to n, say, then you are off by an order of 1/n^2, so getting 100 digits of accuracy would take about 10^50 operations. How does Alpha do it so fast?

Answer 1

I am not sure if this is what Mathematica uses, but I do know that "EulerMaclaurin" is one of the summation acceleration methods it uses.

The Euler-Maclaurin Sum Formula gives asymptotic expansions such as $$ \sum_{k=1}^n\frac1{k^3}=\zeta(3)-\frac1{2n^2}+\frac1{2n^3}-\frac1{4n^4}+\frac1{12n^6}-\frac1{12n^8}+\frac3{20n^{10}}-\frac5{12n^{12}}+O\!\left(\frac1{n^{14}}\right)\tag1 $$ Which leads to the accelerated $$ \zeta(3)=\sum_{k=1}^n\frac1{k^3}+\frac1{2n^2}-\frac1{2n^3}+\frac1{4n^4}-\frac1{12n^6}+\frac1{12n^8}-\frac3{20n^{10}}+\frac5{12n^{12}}+O\!\left(\frac1{n^{14}}\right)\tag2 $$ Applying $(2)$ with $n=100000$, we get $70$ decimals places of $\zeta(3)$: $$ \small1.202056903159594285399738161511449990764986292340498881792271555341838\tag3 $$ Taking more terms of the Euler-Maclaurin expansion or more terms of the sum yieds more precision.