Question

I have the following integral

I wonder how to show that

Below I paste my current attempt in Mathematica.

tmp1 = Sqrt[Sum[(1/i), {i, 1, d}]]
Limit[Integrate[1/Sqrt[(2*\[Pi])] * Exp[-1/2 * z^2], {z, tmp1, Infinity}], d -> Infinity]

Thanks in advance for any pointer or tip.

Answer 1

We can show this by hand. Given we know that the inner function integrates to 1/2 from 0 to infinity we can then say

1/2 = integral from 0 to sqrt(sum( 1/i )) + integral from sqrt(sum(1/i)) to infinity.

Then

1/2 – integral from 0 to sqrt(sum(1/i)) = integral from sqrt(sum(1/i)) to infinity

and since lim d -> infinity of sqrt(sum(1/i)) = infinity we know that the left hand side is zero and this the integral converges to zero.