A college student's dream

Note that the successive the terms in the series rapidly converges to zero. As an estimate, we have $\sum_{n=6}^{\infty} \frac{1}{n^n} \leq \int_{x=5}^{\infty}\frac{1}{x^6} dx = \frac{1}{15625}= 0.000064$ Hence to obtain a result correct upto the third term of the decimal, it suffices to compute the partial sum of the series only up to the first 5 terms, which evaluates to 1.291.

It is clear that the infinite sum converges because : $\displaystyle\sum_{n=1}^{\infty} \frac{1}{n^n} < \displaystyle\sum_{n=1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6} \approx 1.64$ The first 7 terms produces 1.291 to 3 significant figures