11
As I understood you start from the completeness relation
$$\sum_{\ell=0}^\infty \frac{2\ell + 1}{2} P_\ell(x)P_\ell(y) = \delta(x-y)$$
and use that
$$
P_n(0) =
\begin{cases}
\frac{(-1)^{m}}{4^m} \tbinom{2m}{m}
= \frac{(-1)^{m}}{2^{2m}} \frac{(2m)!}{\left(m!\right)^2}
& \text{for} \quad n = 2m
\\
0 & \text{for} \quad n = 2m+1 \,....
8
I think you're misinterpreting Mathematica output. First of all, notice that the first output you get from Mathematica is a message that precludes the subsequent numerical results being sensible (or at the very least must make you very cautious of them):
Sum::div: Sum does not converge.
If you omit the N[] wrapper, you'll get the actual sum simplified to ...
5
V 12.1.1. Just add FullSimplify to help it
FDE[d_, η_] := η^(d + 1)/Gamma[d + 2] + π^2/(6*Gamma[d])*η^(d - 1);
Series[FullSimplify[FDE[d/t, 1/η]/FDE[d/t - 1, 1/η]], {η, 0, 3}]
Using only Simplify gives, on same system the following
Series[Simplify[FDE[d/t, 1/η]/FDE[d/t - 1, 1/η]], {η, 0, 3}]
FullSimplify[%]
3
Recently I wrote a recursive function to calculate the two-point Taylor expansion. You can find it here. The notebook includes a couple of examples that can get you started to play with it.
When I came up with this idea, I must admit that I didn't know the two-point Taylor expansion was a thing (I was playing with polynomial division). So, I didn't follow ...
3
expr = g[f[{a, b}]];
Presumably, the intent is to use rules to replace f and g
expr /. {f -> Total, g -> Cos}
(* Cos[a + b] *)
expr /. {f -> (Apply[Plus, #] &), g -> Cos}
(* Cos[a + b] *)
To get an expansion you need a third function so either f or g must be replaced by a Composition
expr /. {f -> Total, g -> Composition[...
1
The problem seems numerical. Note that the following works fine:
ser = Series[Rationalize[f, 0], {y, 0, 5}]
You could then use N[ser, <yourDesiredPrecision>] for a numerical approximation:
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