Lab07: Sequences & Series

Introduction

In this lab we will use Mathematica to investigate sequences and series.  

First some preliminary notation.  In Mathematica a sequence, or list, can be generated using a Table command.  For example the first 10 term in the harmonic sequence are:

h=Table[1/n,{n,1,10}]

To see the values of this sequence as floating-point numbers (decimal approximations) we could do the following:

N[h]

We could plot the terms in this sequence with the following:

ListPlot[h]

To see if this sequence has a limit, we would need to  investigate the terms for larger values of n.  Observe the following commands:

N[Table[1/n,{n,100,110}]]

N[Table[1/n,{n,1000,1010}]]

N[Table[1/n,{n,10000,10010}]]

Here we see that the hundreth, thousandth and tenthousandth terms appear to be getting closer and closer to zero.

Graphically we could see the following:

ListPlot[Table[{n,1/n},{n,100,110}],PlotRange->{-0.05,0.05}]

ListPlot[Table[{n,1/n},{n,1000,1010}],PlotRange->{-0.05,0.05}]

ListPlot[Table[{n,1/n},{n,10000,10010}],PlotRange->{-0.05,0.05}]

These are the graphs of the above lists of values.  To see a more instructive graph, look at the following:

ListPlot[Table[{n,1/n},{n,1,1000}],PlotRange->{-0.02,0.5}]

For series, we could look at sequences of partial sums.

For example, if we look at (∑_(k=1)^∞)(1/(Sqrt[k])), the first 10 terms of the sequence of partial sums is:

s=Table[Sum[1/Sqrt[k],{k,1,n}],{n,1,10}]

Again, to see the floating-point approximation we would look at:

N[s]

Also the graph of the first 100 terms would be:

ListPlot[Table[{n,Sum[1/Sqrt[k],{k,1,n}]},{n,1,100}]]

This sequence (of partial sums) appears to diverge.  Compare this to the series (∑_(k=1)^∞)((((-1))^(k+1))/(Sqrt[k])):

ListPlot[Table[{n,Sum[(-1)^(k+1)/Sqrt[k],{k,1,n}]},{n,1,100}]]

This sequence (of partial sums) does appear to be converging to about 0.6.

Exercises

1.  Consider the sequence given by (Subscript[a, n])=((Sin[n])/(Sqrt[n])), for n = 1, 2, 3, ...

a.  List the decimal approximations of the first ten term of this sequence.

b.  List the decimal approximations of terms (Subscript[a, 100]) through (Subscript[a, 110]).

c.  Generate an appropriate graph of this sequence and determine if the sequence converges or diverges.  If it converges, find it’s limit.

2.  Consider the sequence given by (Subscript[a, n])=2+(((1.001))^n).  Do parts a, b, and c from above for this sequence.

3.  Consider the sequence given by (Subscript[a, n])=((Sqrt[(n^2)+2])/(2n+101)), n = 1, 2, 3, ...  Do parts a, b, and c from above for this sequence.

4.  Consider the series (∑_(k=1)^∞)(((2^k)k!)/(k^k)).  For the sequence of partial sums, do parts a, and b from above for this sequence.  Then determin if the series converges or diverges.  If it converges, exstimate it’s value.  (Note that is may take a long time to calculate the (n^th) partial sum for n > 150.)