I am trying to understand why Sum does not recognize a particular Taylor series as convergent. I have defined a function 'series' like this, that computes the Taylor series of a function 'f' at 'x', expanded around 'a' up to 't' terms. It seems to work for many functions.
In[1]:= series[f_, x_, a_, t_] := Sum[((x - a)^k*Derivative[k][f][a])/k!, {k, 0, t}]
In[2]:= series[Cos, 4, 3, Infinity]
Out[2]= Cos[4]
In[20]:= series[Log, 4, 3, Infinity] // PowerExpand // Simplify
Out[20]= Log[4]
However, it doesn't seem to work for the reciprocal function. It looks right for the first several partial sums:
In[4]:= f[x_] := 1/x
In[5]:= Table[series[f, 4.`20, 3, u], {u, 1, 20}]
Out[5]= {0.22222222222222222222, 0.25925925925925925926, 0.24691358024691358025, \
0.25102880658436213992, 0.24965706447187928669, 0.25011431184270690444, \
0.24996189605243103185, 0.25001270131585632272, 0.24999576622804789243, \
0.25000141125731736919, 0.24999952958089421027, 0.25000015680636859658, \
0.24999994773121046781, 0.25000001742292984406, 0.24999999419235671865, \
0.25000000193588109378, 0.24999999935470630207, 0.25000000021509789931, \
0.24999999992830070023, 0.25000000002389976659}
But it chokes on symbolic infinity.
In[6]:= series[f, 4, 3, Infinity] // InputForm
During evaluation of In[6]:= Sum::div: Sum does not converge.
Out[6]//InputForm=
Sum[(3^(-1 - k)*FactorialPower[-1, k])/k!, {k, 0, Infinity}]
I don't really understand why, because it seems to me that these forms should be equivalent:
In[7]:= FactorialPower[n, k]/(k!) == Binomial[n, k] // FunctionExpand
Out[7]= True
In[8]:= Table[FactorialPower[-1, k]/(k!), {k, 0, 10}]
% ==
Table[Binomial[-1, k], {k, 0, 10}] ==
Table[(-1)^k, {k, 0, 10}]
Out[8]= {1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1}
Out[9]= True
And when I use these forms in Sum, there's no issue.
In[10]:= Sum[3^(-1 - k)*Binomial[-1, k], {k, 0, Infinity}] // InputForm
Out[10]//InputForm=
1/4
In[11]:= Sum[3^(-1 - k)*(-1)^k, {k, 0, Infinity}] // InputForm
Out[11]//InputForm=
1/4
I am wondering if there might be a bug in SumConvergence? I observe the following:
In[12]:= SumConvergence[3^(-1 - k)*FactorialPower[-1, k]/(k!), k]
Out[12]= False
In[13]:= SumConvergence[3^(-1 - k)*Binomial[-1, k], k]
Out[13]= True
In[14]:= SumConvergence[3^(-1 - k)*(-1)^k, k]
Out[14]= True
I am using Mathematica 13.3 on Windows 11.
