The integral expression  converges.

By the definition of improper integrals of type 1:

Substitute above result in the expression (1).

Apply limit chain rule :

If  ,    and  is continuous at , then .

So in order to calculate , we must calculate 

If , then above expression tends to .

If , then expression tends to .

If , then above expression tends to a finite number .

The limit value is converges when  only otherwise it is diverges.

When ,

Therefore, .

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