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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.

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1 Answer 1

1
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Indeed,

FactorialPower[n, k]/(k!) == Binomial[n, k] // FunctionExpand

True

However, the above result is only generic:

n = -1; FactorialPower[n, k]/(k!) == Binomial[n, k] // FunctionExpand

ComplexInfinity == ComplexInfinity

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  • $\begingroup$ FunctionDomain[Binomial[n, k], {n, k}] results in k \[Element] Integers || n >= 0 || n \[NotElement] Integers. $\endgroup$
    – user64494
    Apr 7 at 6:50
  • $\begingroup$ And yet, Table[FactorialPower[-1, k]/(k!), {k, 0, 10}]== Table[Binomial[-1, k], {k, 0, 10}] == Table[(-1)^k, {k, 0, 10}], showing that these are equivalent at least for nonnegative K, which is what's happening in the Sum. So why does Sum reject it? Again, numerically one can see that it converges, as one would expect. The function in question is theoretically analytic at that point. $\endgroup$ Apr 7 at 15:52
  • $\begingroup$ @GlennWelch: The left hand FunctionDomain[Binomial[n, k], {n, k}] does not know what the right hand Table[Binomial[-1, k], {k, 0, 10}] does. $\endgroup$
    – user64494
    Apr 7 at 17:16

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