Chapter 8: Infinite Sequences and Series

Section 8.3: Convergence Tests

Example 8.3.12

$$ Determine if the series $\sum _{n\=1}^{\infty }1\/n^n$ diverges, converges absolutely, or converges conditionally. If it converges conditionally, determine if it also converge absolutely. $$ Solution Mathematical Solution $$ Applying the Ratio test, compute    $\underset{n\to \infty }{lim}\frac{a_{n\+1}}{a_n}\=\underset{n\to \infty }{lim}\frac{\left(\frac{1}{{\left(n\+1\right)}^{n\+1}}\right)}{\left(\frac{1}{n^n}\right)}\=\underset{n\to \infty }{lim}{\left(\frac{n}{n\+1}\right)}^n\frac{1}{n\+1}\=0$    from which it follows that the given series converges absolutely. $$ Maple Solution $$ The most efficient way to implement the Ratio test in Maple, is to define the general term $a_n$ as a function $a\left(n\right)$.   •  Write $a\left(n\right)\=\dots$ •  Context Panel: Assign Function $a\left(n\right)\=\frac{1}{n^n}$$\stackrel{\text{assign as function}}{\to }$$a$ •  Calculus palette: Limit operator •  Context Panel: Evaluate and Display Inline $\underset{n\to \infty }{lim}\frac{a\left(n\+1\right)}{a\left(n\right)}$ = $0$$$ $$

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