Basic Mathematics20775 marksNumerical: series convergenceIntegral test
Determine the convergence or divergence of the series sumn=1infty n2 e-n.
Determine the convergence or divergence of the series $\sum_{n=1}^{\infty} n^2 e^{-n}$. [5]
Series: $\displaystyle\sum{n=1}^{\infty} n^2 e^{-n}$ General term: $an = n^2 e^{-n} = \dfrac{n^2}{e^n}$ All terms are positive, so tests for positive-term series (Ratio Test, Root Test) apply. For a positive series, compute $$L = \lim{n \to \infty} \frac{a{n+1}}{an}.$$ $$\frac{a{n+1}}{an} = \frac...