SequenceLimit became NumericalMath``NSequenceLimit. In the past, that change broke some of the examples on NLimit that used SequenceLimit. I reported the breakage, and the examples were fixed to use the same syntax but not actually call the now-non-existent function. The only remaining reference in the documentation that I can find to the use of SequenceLimit as if it were still a function is on the last bullet point of NLimit's detailed description for its Method option: "uses SequenceLimit on constructed sequence".
Recently, I started investigating the behavior of NumericalMath``NSequenceLimit because I need it in my engineering work. I feel NumericalMath``NSequenceLimit has some quite confusing shortcomings in regard to the approximation it returns vs. the number of terms supplied, so I wrote my own version from some references.
A comparison appears below showing the convergence of partial sums of series approximations to Pi and E. Afterward, I use symbolic lists to document several parts of NumericalMath``NSequenceLimit that seem... possibly wrong... to me.
sPi@n_:=Sum[4(-1)^nn/(2nn+1),{nn,0,n}];
Table[sPi@n,{n,0,4-1}]//N
(*{4.,2.66667,3.46667,2.89524}*)
NumericalMath`NSequenceLimit@%
(*3.16667*)
sequenceLimit@%%
(*3.13333*)
(*my answer is closer to Pi~=3.141, for good reason as shown
below in the general case with 4 terms {a,b,c,d}*)
sE@n_:=Piecewise[{{(1+1/n)^n ,n!=0}},1]
Table[sE@n,{n,0,4-1}]//N
(*{1.,2.,2.25,2.37037}*)
NumericalMath`NSequenceLimit@%
(*2.33333*)
sequenceLimit@%%
(*2.48214*)
(*my answer is closer to E~=2.71828, for good reason as shown
below in the general case with 4 terms {a,b,c,d}*)
Below, I illustrate the difference between the behavior of the built-in NumericalMath``NSequenceLimit function vs. my sequenceLimit function for symbolic lists, like {a,b,c,d}.
Quiet@NumericalMath`NSequenceLimit[{a,b,c}]
(*NumericalMath`NSequenceLimit[{a,b,c}]*)
(*yes, unevaluated with 3 inputs*)
sequenceLimit[{a,b,c}]//Simplify (*my function*)
(*(-b^2+a*c)/(a-2*b+c)*)
(*same as Mathematica's next result with 4 inputs*)
NumericalMath`NSequenceLimit[{a,b,c,d}]
(*(-b^2+a*c)/(a-2*b+c)*)
(*yes, only references a, b, & c not the later, more converged, d*)
sequenceLimit[{a,b,c,d}]//Simplify (*my function*)
(-c^2+b*d)/(b-2*c+d)
(*note use of d, which explains why my approximations
for Pi and E above are better*)
Quiet@NumericalMath`NSequenceLimit[{a,b,c,d,e}]//Simplify
(*(-b^2+a*c)/(a-2*b+c)*)
(*yes, same result as with 4 terms, no d or e in output*)
sequenceLimit[{a,b,c,d,e}]//Simplify (*my function*)
(*(c^3+a*d^2+b^2*e-c*(2*b*d+a*e))/
(b^2+3*c^2-2*c*d+d^2-2*b*(c+d-e)-c*e-a*(c-2*d+e))*)
(*note use of d and e*)
I also document some error messages that seem spurious.
NumericalMath`NSequenceLimit[{a}]
(*NumericalMath`NSequenceLimit::seqw Sequence of length 1
is too short for use with Degree -> 1*)
(*NumericalMath`NSequenceLimit::bdmtd WynnEpsilon is not
a valid specification of a sequence limit extrapolation
algorithm.*)
(*NumericalMath`NSequenceLimit[{a}]*)
NumericalMath`NSequenceLimit[{a},Degree->1]; (*Nulled output*)
(*NumericalMath`NSequenceLimit:optx Unknown option ° in
NumericalMath`NSequenceLimit[{a}]*)
NumericalMath`NSequenceLimit[{a},"Degree"->1]; (*Note quotes*)
(*NumericalMath`NSequenceLimit::optx Unknown option "Degree"
in NumericalMath`NSequenceLimit[{a}]*)
sequenceLimit[{a}] (*sequence limit is my function*)
(*a*)
Quiet@NumericalMath`NSequenceLimit[{a,b}]
(*NumericalMath`NSequenceLimit[{a,b}]*)
sequenceLimit[{a,b}] (*my function*)
(*b*)
Everything above is from the current version of Mathematica Online.
$Version
11.3.0 for Linux x86 (64-bit) (March 7, 2018)
Here is the definition of my function
wynnE[-2,_,_]:=0
wynnE[-1,n_,s_]:=s@n
wynnE[rkp1_,n_,s_]:=(*wynnE[rkp1,n,s]=*)(*optional caching*)
wynnE[rkp1-2,n+1,s]+1/(wynnE[rkp1-1,n+1,s]-wynnE[rkp1-1,n,s])
sequenceLimit[list_?VectorQ]:=
With[{len=Length@list},
With[{off=Boole@EvenQ@len},
wynnE[len-2-off,off,list[[#+1]]&]
]/;len>0
]