Some families of rapidly convergent series representations for the zeta functions (Q5928409)

This valuable historical survey describes various families of rapidly convergent series for evaluating the Riemann zeta function at odd integers \(\geq 3\). One striking new result is the series NEWLINE\[NEWLINE\zeta(3)=-{6 \pi^2 \over 23}\sum^\infty_{k=0} {(98k+121) \zeta(2k) \over(2k+1) (2k+2)(2k+3) (2k+4) (2k+5)2^{2k}}NEWLINE\]NEWLINE which converges more rapidly than Euler's famous series NEWLINE\[NEWLINE\zeta(3) =-{4\pi^2\over 7}\sum^\infty_{k=0} {\zeta(2k) \over(2k+1) (2k+2) 2^{2k}},NEWLINE\]NEWLINE or Apéry's, NEWLINE\[NEWLINE\zeta(3)= {5\over 2}\sum^\infty_{k=1} {(-1)^{k-1} \over k^3{2k \choose k}}.NEWLINE\]