Studying the behavior of the series $$ \sum _{n=1}^{\infty } \left(1-\frac{\log (n)}{n}\right)^{2 n}$$, I try in 12.3.1 on Windows 10

SumConvergence[(1 - Log[n]/n)^(2 n), n]

False

, but

SumConvergence[(1 - Log[n]/n)^(2 n), n, Method -> "RaabeTest"]

True

The result of

NSum[(1 - Log[n]/n)^(2 n), {n, 1, Infinity}]

1.33193*10^244

confirms the divergence and the results of

NSum[(1 - Log[n]/n)^(2 n), {n, 1, Infinity}, WorkingPrecision -> 30]

1.407104427435176587354

and

Series[(1 - Log[n]/n)^(2 n), {n, Infinity, 3}]

(1/n)^2-Log[n]^2/n^3+O(1/n)^4

stand for the convergence.

What should I trust?