Limit of Taylor coefficients

Question

I want to calculate the following limit:

b[n_] := (D[1/(1 - t), {t, n}] /. {t -> 0})/Factorial[n]
Limit[b[n], n -> ∞]
(* Indeterminate *)

where b[n] represents the $n^\rm{th}$ Taylor coefficient of function $1/(1-t)$ at $t=0$.

1. Each $b_n$ is 1. Why does the limit return Indeterminate and not 1?

2. Is there any simple method to calculate the limit of Taylor coefficients of a rational function? Below is an example of a rational function:

(1 + t)/(-t (1 + t + t^3) + (1 + t) (1 - t^2)^3 (1 - t^4)^3 (1 - t^6)^3)

Any help is highly appreciated. Thanks is advance.

Answer 1

Use DiscreteLimit.

f = 1/(1 - t);
b = (D[1/(1 - t), {t, n}] /. {t -> 0})/Factorial[n]
(* ((-1)^n FactorialPower[-1, n])/n! *)

DiscreteLimit[b, n -> ∞]
(* 1 *)

Also, you don't need to calculate the coefficient via D and Factorial because there exists SeriesCoefficient.

b = SeriesCoefficient[f, {t, 0, n}]
(* Piecewise[{{1, n >= 0}}, 0] *)

DiscreteLimit[b, n -> ∞]
(* 1 *)

Now to your complicated rational function, for which it seems that the coefficients keep growing indefinitely.

f = (1 + t)/(-t (1 + t + t^3) + (1 + 
       t) (1 - t^2)^3 (1 - t^4)^3 (1 - t^6)^3);

Table[SeriesCoefficient[f, {t, 0, n}], {n, 0, 25}]
(* {1, 1, 4, 7, 20, 41, 100, 221, 515, 1164, 2679, 6096, 13980, 31889, 
    73002, 166725, 381369, 871473, 1992675, 4554587, 10412772, 23802307,
    54414141, 124388286, 284356379, 650034525} *)

You can obtain a general formula, which is an extremely long expression with 37 terms and many Root objects.

b = SeriesCoefficient[f, {t, 0, n}]

You may try using Limit or DiscreteLimit on this result if you are patient. However, with a bit dangerous approach by looking at the numerical representation, you may see that there are only exponential terms, and the largest base (by absolute value) is positive.

Collect[N[b], n] /. a_Complex :> Abs[a]
(* 0.225774 * (-1.4473)^n + ... + 0.685494 * 2.28601^n *)

Limit[N[b], n -> ∞]
∞