# How to study convergence of recurrence relations?

Is there a way to test the convergence of a recurrence relation in Mathematica?

For example does this relation converge:

$$a_{n+1}=a_n \dfrac{n-\dfrac{1}{4}+c^2}{-5+5c^2}$$

for some values of the parameter $$c$$?

• You could try solving it first: RSolve[{a[n + 1] == a[n] (n - 1/4 + c^2)/(5 c^2 - 5), a == a0}, a,n] - which gives a complicated solution involving the Pochammer function. Then use Limit like so: Limit[1/4 5^-n a0 (-1 + c^2)^-n (-1 + 4 c^2) Pochhammer[3/4 + c^2, -1 + n], n -> Infinity] which gives ComplexInfinity. So it doesn't converge for arbitrary a0 and c, except trivially at a0 = 0. Jul 23 '20 at 16:02
• If you reverse the limits, it converges to zero. Defining seq[c_] := RSolve[{a[n] == a[n - 1] (n - 1/4 + c^2)/(5 (c^2 - 1)), a == a0}, a[n], n], then Limit[a[n] /. seq[c], c -> Infinity] gives {5^-n a0}, so it vanishes for n->Infinity. Not super interesting, but I thought I might as well point it out. Jul 23 '20 at 16:15
• Oh, wow, it took me way too long to realize that that immediately follows from the recurrence relation. Jul 23 '20 at 16:37
• For very large $n$ you have approximately $a_{n+1}\approx d\cdot a_n n$ for $d=1/(5c^2-5)$, and therefore approximately $a_n\propto d^n \Gamma(n)$, which diverges strongly $\forall c\in\mathbb{R}$. Jul 23 '20 at 17:10