Chapter 8: Infinite Sequences and Series
Section 8.3: Convergence Tests
Determine if the series ∑n=0∞e−nn! diverges, converges absolutely, or converges conditionally.
If it converges conditionally, determine if it also converge absolutely.
Since limn→∞e−n n=∞, by the nth-term test, the given series must diverge because an=e−n n! does not tend to zero as n→∞.
The following two Maple calculations determine the divergence of the given series. In the first, Maple sums the series and obtains infinity as the result. In the second, Maple shows that the nth-term does not approach zero as n→∞. Either of these results suffices to establish the divergence of the series, although the first is not as convincing as the second because the Maple process behind the summation of the series is not exposed to inspection.
Expression palette: Summation template
Context Panel: Evaluate and Display Inline
∑n=0∞ⅇ−n n! = ∞
Calculus palette: Limit operator
limn→∞ⅇ−n n! = ∞
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