不知名的碎片 1

不知名的碎片 1

solution 1:

Let

Note that our sum equals

Now decreases from to on Thus, for

Summing on then gives

This is true for any As the integral on the left approaches By the squeeze theorem, the limit of our sum equals the value of this integral, which is

solution 2:

Denote. We have

Weierstrass factorization for hyperbolic sine is:

Therefore

And finally taking the limit as

solution 3:

Using partial fraction decomposition and summing

Using the asymptotic of generalized harmonic numbers

Combining all the above

另一个碎片