Evaluate infinite sums and products

Sum[1 / n ^ 2, {n, 1, Infinity}]

Pi^2/6

Sum[1 / n ^ 2, {n, 1, 100}]

1589508694133037873112297928517553859702383498543709859889432834803818131090369901/972186144434381030589657976672623144161975583995746241782720354705517986165248000

Sum[1. / n ^ 2, {n, 1, 100}]

1.6349839001848923

Limit[Abs[Pi ^ 2 / 6 - Sum[1 / n ^ 2, {n, 1, k}]], k -> Infinity]

0

Sum[1 / n ^ s, {n, 1, Infinity}]

Zeta[s]

Pi z Product[1 - z ^ 2 / n ^ 2, {n, 1, Infinity}]

Pi*z*Product[1 - z^2/n^2, {n, 1, Infinity}]

Product[1 / (1 - Prime[k] ^ -s), {k, 1, Infinity}]

Product[(1 - Prime[k]^(-s))^(-1), {k, 1, Infinity}]