p. 649

41.

a)

s[k_]  :=  (∑_(n = 1)^k) (1.0/((n^3) Sin[n]^2))

Limit[s[k], k -> ∞]

b)

z  =  Table[s[k],  {k, 1, 100}];

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

s[100]

The limit looks like it might be approaching a number close to 4.8.

c)

z  =  Table[s[k],  {k, 1, 200}];

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

s[200]

The plot of the first 200 points seems to confirm the original guess.

d)

z  =  Table[s[k],  {k, 1, 400}];

ListPlot[z, PlotRange -> {0, 30}]

s[354]

s[355]

The sequence jumps from (Subscript[s, 354]) = 4.8 to (Subscript[s, 355]) = 29.4!

N[355/113]

Since 355  ≃  113  π, \(TraditionalForm
      \`sin(355) ≃ \ sin(113  π)\  = \ 0\) and the 355th term of the series suddenly becomes very large compared to nearby terms:

(1. /((354^3) Sin[354]^2))(1. /((355^3) Sin[355]^2))(1. /((356^3) Sin[356]^2))

We should expect to see a jump in (Subscript[s, n]) whenever n is close enough to a multiple of π to make (n^3) sin(n)a small number.

Do[If [
        FractionalPart[N[ n π ]] <  .001 || 
          FractionalPart[N[ n π ]] >  .999, Print[ N[n π]]], {n, 
        100, 500}]

Jumps should occur when n = 710, n = 1065, and n = 1420.

s[708]s[709]s[710]s[711]

s[1063]s[1064]s[1065]s[1066]

s[1418]s[1419]s[1420]s[1421]

The jumps are small, but very noticible compared to how little the nearby partial sums are changing. Such jumps make it difficult to assess whether the series will ultimately converge. We do not have an easy way to estimate how small sin(n) may become (it is never zero of course), and therefore, no easy way to estimate how large (1/((n^3) (sin^2)(n))) may become. The function NSum also has trouble with this series:

NSum[(1.0/((n^3) Sin[n]^2)), {n, 1, ∞}]