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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