Summation to ∞ and beyond

We know that the summation sequence 1/1+1/2+1/3+1/4… goes to ∞

And the summation sequence 1/1+1/4+1/9+1/16 … = π^2/6

So there is some x beyond which 1^(-x) + 2^(-x) + 3^(-x) … sums to a finite number and beneath which the sum tends to ∞

Does anyone have an idea on what x is, and if possible a proof

Note: According to the Riemann zeta function 1^(-1) + 2^(-1) … goes to -1/12, so my question in that context would be when would the Riemann zeta function give the output as ∞/-∞ , not sure but I think both of these questions are the same ( if not please tell )

Easy Math Editor

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Comments

$x>1$ gives a convergent sum. $x\leqslant 1$ gives a divergent sum.

Look up Convergence test for $p$ -series.

Pi Han Goh - 4 months, 3 weeks ago

I looked it up right now and it says it’s at 1?!