Examples for Section 8.6
The Ratio and Root Tests for Series

p. 654

1.

(∑_(n = 1)^∞)((n^((Sqrt[2])))/(2^n)). Let us try the ratio test: (lim_(n -> ∞)) (Subscript[a, n + 1])/(Subscript[a, n])=(lim_(n -> ∞))((((n + 1))^((Sqrt[2])))/(2^(n + 1)))/((n^((Sqrt[2])))/(2^n))=(lim_(n -> ∞))(1/2)(((1 + (1/n)))^((Sqrt[2])))=(1/2).
Since the limit is less than 1, the series converges.

Limit[((((n + 1))^((Sqrt[2])))/(2^(n + 1)))/((n^((Sqrt[2])))/(2^n)), 
      n -> ∞]

NSum[((n^((Sqrt[2])))/(2^n)), {n, 1, ∞}]

9.

\(TraditionalForm
      \`∑\+\(n = 1\)\%∞\((1\  - \ 3\/n)\)\^n\). The root test gives: \(TraditionalForm
      \`\(lim\_\(n -> ∞\)\((a\_n)\)\^\(1/n\) = 
        \(lim\_\(n -> ∞\)\((1\  - \ 3\/n)\) = 1\)\ \)\) which is inconclusive. However, we should note that the terms of the series do not approach 0: (lim_(n -> ∞))(((1 - (3/n)))^n) = (e^((lim_(n -> ∞))n ln(1 - (3/n))))=(e^(-3))(by L'Hopital's Rule). Therefore, the series diverges.

Limit[(((1 - (3/n)))^n), n -> ∞]

11.

(∑_(n = 1)^∞)(( ln(n))/(3^n)). The root test reveals: \(TraditionalForm
      \`lim\_\(n -> ∞\)\((\(ln(n)\)\/3\^n)\)\^\(1/n\) = 
        \(\(1\/3\) \(lim\_\(n -> ∞\)\((ln(n))\)\^\(1/n\)\) = 
          \(\(1\/3\) 
              \(lim\_\(n -> ∞\)e\^\(\(ln(ln(n))\)/n\)\) = 
            1\/3\)\)\)(by L'Hopital's Rule). So the series converges.

Limit[(1/3) Log[n]^(1/n), n -> ∞]

NSum[Log[n]/(3^n), {n, 1, ∞}]

27.

Consider the series (∑_(n = 1)^∞) (Subscript[a, n]) whose terms are defined recursively by (Subscript[a, 1]) = 2, (Subscript[a, n + 1]) =  ((1 + sin(n))/n) (Subscript[a, n]). The ratio test provides an easy method to determine convergence: \(TraditionalForm
      \`lim\_\(n -> ∞\)a\_\(n + 1\)/a\_n = \ 
        \(lim\_\(n -> ∞\)\(1 + sin(n)\)\/n = 0\)\) . Therefore, the series converges.

a[1] = 2.0;  a[n_]  :=  ((1.0 + Sin[n])/n) a[n - 1]

s[k_]  :=  (∑_(n = 1)^k) a[n]

z  =  Table[s[k], {k, 1, 15}];

ListPlot[z, PlotRange -> {{0, 15}, {0, 5}}]

s[10]

s[100]

The series converges to a number near 4.68012.