Sums over Bijections

Algebra Level 5

Does there exist a bijective function $f$ from the positive integers onto itself, sucht that ...

$(I)$ ... the sum $\sum _{ n=1 }^{ \infty }{ \frac { 1 }{ f\left( n \right) +f^{ -1 }\left( n \right) } }$ converges?

$(II)$ ... the sum $\sum _{ n=1 }^{ \infty }{ \frac { { (-1) }^{ n } }{ f\left( n \right) +f^{ -1 }\left( n \right) } }$ diverges?

$(III)$ ... the sum $\sum _{ n=1 }^{ \infty }{ \frac { 1 }{ f\left( n \right) *f^{ -1 }\left( n \right) } }$ diverges?

Note: $f^{ -1 }\left( n \right)$ is the inverse function of $f\left( n \right)$ .

Only

(II)

and

(III)

are true Only

(I)

and

(III)

are true Only

(I)

is true

(I)

(II)

and

(III)

are true None of them are true Only

(I)

and

(II)

are true Only

(III)

is true Only

(II)

is true

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