Mathematics I20745 marksNumerical: series convergence testConvergence tests and power series
Test the convergence of the series sumn=1infty fracnnn!
Test the convergence of the series $\sum_{n=1}^{\infty} \frac{n^n}{n!}$ [5]
Series to test: $$\sum{n=1}^{\infty} un, \qquad un = \frac{n^n}{n!}$$ Because the general term involves a factorial $n!$, D'Alembert's Ratio Test is most convenient. $$un = \frac{n^n}{n!}, \qquad u{n+1} = \frac{(n+1)^{n+1}}{(n+1)!}$$ Form the ratio: $$\frac{u{n+1}}{un} = \frac{(n+1)^{n+1}}{(n+1)!...