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I was playing around with an infinite series recently and I noticed something peculiar, I was hoping somebody could clarify something for me.

Suppose we have an infinite series of the form:

[itex]\sum^_{n = 1}^{\infty} 1/n^\phi[/itex]

where [itex]\phi[/itex] is some even natural number, it appears that it is always convergent to a rational multiple of [itex]\pi[/itex].

Now if we take this series and change it slightly:

[itex]\sum^_{n = 1}^{\infty} 1/n^\alpha[/itex]

where [itex]\alpha[/itex] is some even natural number, it appears that it is always convergent to the Riemann Zeta Function evaluated at [itex]\alpha[/itex] i.e. [itex]\zeta(\alpha)[/itex].

Can someone explain the relationship expressed in these infinite series?

EDIT: What's wrong with my LaTeX for the infinite series?

Suppose we have an infinite series of the form:

[itex]\sum^_{n = 1}^{\infty} 1/n^\phi[/itex]

where [itex]\phi[/itex] is some even natural number, it appears that it is always convergent to a rational multiple of [itex]\pi[/itex].

Now if we take this series and change it slightly:

[itex]\sum^_{n = 1}^{\infty} 1/n^\alpha[/itex]

where [itex]\alpha[/itex] is some even natural number, it appears that it is always convergent to the Riemann Zeta Function evaluated at [itex]\alpha[/itex] i.e. [itex]\zeta(\alpha)[/itex].

Can someone explain the relationship expressed in these infinite series?

EDIT: What's wrong with my LaTeX for the infinite series?

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