I Does the sum of all series 1/n^m, m>1 converge?

[BWV]

$\sum_{n=1}^\infty 1/n^2$ converges to $π^2/6$

and every other series with n to a power greater than 1 for n∈ℕ convergesis it known if the sum of all these series - $\sum_{m=2}^\infty \sum_{n=1}^\infty 1/n^m$ for n∈ℕ converges?

apologies for any notational flaws

 

[BWV]

Duh, so as every sum is >1 they of course the double sum above would diverge, but is there anything interesting about the partial sums or

$\sum_{m=2}^\infty \sum_{n=2}^\infty 1/n^m$ ?

 

[mathman]

A more interesting question is what is the result when the n sum starts at n=2.

 

[BWV]

A more interesting question is what is the result when the n sum starts at n=2.

Looks like it converges to 1

ran n to 2:1000 then 2:10000 in Matlab, n(1000)=0.9990, n(10,000)=0.9999

 

[TeethWhitener]

Duh, so as every sum is >1 they of course the double sum above would diverge, but is there anything interesting about the partial sums or

$\sum_{m=2}^\infty \sum_{n=2}^\infty 1/n^m$ ?

Flip the sums and focus on the $\sum_{m=2}^\infty 1/n^m$ part first.