Section 1.3: order of convergence

Andy Long, Spring, 2024

To answer Danny’s question, about how we know (or at least suspect) that, for the sequence of 1a, α =1:

a[n_]:=(n!)/(n^2)
alpha[a_,p_,n_]:=Log[Abs[(a[n+2]-p)/(a[n+1]-p)]]/Log[Abs[(a[n+1]-p)/(a[n+0]-p)]]
TableForm[Table[{n,N[alpha[a,0,n]]},{n,1,20}]]

Here’s a sequence that converges to 0, with order > 1:

b[n_]:=2^(1-2^n)
TableForm[Table[{n,N[alpha[b,0,n]]},{n,1,10}]]

Last time we were looking at the order of convergence of those sequences in Table 1.4:

Let’s have another look, and see why we were getting some strange results!

qval={3,2.9436563656918,2.89858145824525,2.86252153228801,2.83367359152222,2.81059523890958,2.79213255681947,2.77736241114739,2.76554629460972,2.75609340137958,2.74853108679547};
rval={3,2.86799618929986,2.78315514435127,2.73974041668143,2.72324781752852,2.71899828870116,2.71833715075158,2.71828369688657,2.71828184959225,2.71828182851528,2.71828182845907};
sval={3,2.82129001274358,2.73850656616954,2.71973377603211,2.71830229432561,2.71828184916891,2.71828182845934,2.71828182845904,2.71828182845904,2.71828182845904,2.71828182845904};
tval={3,2.78177393100014,2.72150682612711,2.71829014894701,2.71828182851442,2.71828182845904,2.71828182845904,2.71828182845904,2.71828182845904,2.71828182845904,2.71828182845904};

testSequence[val_,times_]:=Module[{},
seq[n_]:=val[[n]];
NumberForm[TableForm[Table[{
n,N[alpha[seq,E,n]]
},{n,1,times}]],16]
]

testSequence[tval,4]
testSequence[sval,6]
testSequence[rval,9]
testSequence[qval,9]