Solving recurrence relations with negative powers or reciprocals

Question

What are the methods that can be used to solve recurrence relations such as,

and reduce $a_n$ to a closed-form formula? And are there any general ways to solve this for arbitrary negative powers like $a_{n+1}=a_n+a_n^{-k}$ ?

something related was asked some hours ago but didn't have much tracking, and there is a look at the asymptotics of this sequence here and a try to a closed formula here — Dabed, Jul 01 '19 at 09:19

score 0 · Accepted Answer · edited Jun 12 '20 at 10:38

0

Taking the first one $$a_{n+1}=a_n+\frac{1}{a_n}\qquad \text{with} \qquad a_1=1$$ have a look here.

No closed form (neither asymptotics).

edited Jun 12 '20 at 10:38

Community

answered Jul 01 '19 at 08:35

Claude Leibovici

Although, there are asymptotics to this sequence as shown in the comments – Peter Foreman Jul 01 '19 at 10:03

1 Answers1