Prove Lim n - > infinite n ^ n / (n!) ^ 2 = 0 by using necessary conditions of series convergence Please

Prove Lim n - > infinite n ^ n / (n!) ^ 2 = 0 by using necessary conditions of series convergence Please

Lim n - > infinite n ^ n / (n!) ^ 2
=Lim n - > infinite Π (I = 1 → n) [n / (I & #178;)]
=Lim n - > infinite e ^ ln [Π (I = 1 → n) n / (I & # 178;)]
=Lim n - > infinite e ^ ∑ (I = 1 → n) ln 1 / [n · (I / N) & #178;]
=Lim n - > infinite e ^ ∑ (I = 1 → n) - n · (1 / N) · [ln n + ln (I / N) & # 178;]
=Lim n - > infinite e ^ {- n · (LN n) - ∑ (I = 1 → n) n · ln (I / N) & # 178; · (1 / N)}
=Lim n - > infinite e ^ {- n · (LN n) - n ·∫ ln X & # 178; DX}
=Lim n - > infinite e ^ {- n · (LN n) - n · [x · ln X & # 178; | - ∫ x D ln X & # 178;]}
=Lim n - > infinite e ^ {- n · (LN n) - n · [0 - ∫ x · (2x) / X & # 178; DX]}
=Lim n - > infinite e ^ {- n · (LN n) - n · [- 2 ∫ DX]}
=Lim n - > infinite e ^ {- n · [(LN n) - 2]}
When Lim n - > infinite, (LN n) - 2 → infinite
Then - n · [(LN n) - 2] → - ∞
Therefore, the original limit = Lim n - > infinite e ^ {- n · [(LN n) - 2]} = 0