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Calculus Level 5

\[\zeta ({ s }_{ 1 },\ldots ,{ s }_{ k })=\sum _{ 0<{ n }_{ 1 }<{ n }_{ 2 }<\cdots <{ n }_{ k } }^{ }{ \frac { 1 }{ { n }_{ 1 }^{ { s }_{ 1 } }{ n }_{ 2 }^{ { s }_{ 2 } }\cdots { n }_{ k }^{ { s }_{ k } } } } \]

Define the multiple zeta function as shown above. Then find

\[\lim _{ s_{ 2 }\rightarrow 0 }{ \lim _{ { s }_{ 1 }\rightarrow 0 }{ \zeta (s_{ 1 },{ s }_{ 2 }) } } +\lim _{ { s }_{ 1 }\rightarrow 0 }{ \lim _{ { s }_{ 2 }\rightarrow 0 }{ \zeta ({ s }_{ 1 },{ s }_{ 2 }) } } . \]

Note:

The multiple zeta function in the limit is defined by its analytic continuation.

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