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?