[Math] Find the radius of convergence given a recurrence relation.

Question

I have the following recurrence relation for the coefficients $c_n$ of the power series: $y = \sum^\infty_{n=0}c_nx^n$

$$c_{n+1} = \frac{n+2}{5(n+1)}c_n$$

What is the radius of convergence of the power series?

I can find the radius of convergence by finding a formula for $c_n$ and then find the limit of $|\frac{c_n}{c_{n+1}}|$ as $n$ approaches $\infty$.

Is there a way to determine its radius of convergence without finding the formula first?

Answer 1

This may be a sledgehammer for your problem but there is a general theorem.

Poincare theorem$\color{blue}{{}^{[1]}}$

Given any recurrence relations with non-constant coefficients $$x_{n+k} + p_1(n)x_{n+k-1} + p_2(n)x_{n+k-2} + \cdots + p_k(n) x_{n} = 0\tag{*1}$$ such that there are real numbers $p_i, 1 \le i \le k$ with $$\lim_{n\to\infty} p_i(n) = p_i, \quad 1 \le i \le k$$ and the roots $\lambda_1, \lambda_2 \ldots, \lambda_k$ for the associated characteristic equation: $$\lambda^k + p_1 \lambda^{k-1} + \cdots + p_k = 0$$ have distinct moduli.

For any solution of $(*1)$, either $x_n = 0$ for all large $n$ or $\displaystyle\;\lim_{n\to\infty} \frac{x_{n+1}}{x_n} = \lambda_i\;$ for some $i$.

In short, if the coefficients of a recurrence relations converges and the corresponds $|\lambda_i|$ are distinct, then either the sequence $x_n$ terminates (i.e. infinite radius of convergence) or the radius of convergence is one of $\frac{1}{|\lambda_i|}$.

For your case, it is clear $c_n$ didn't terminate. Since the characteristic equation of your sequence "converge" to $\lambda - \frac15$, its radius of convergence is $5$.

Notes/References

• $\color{blue}{[1]}$ - Saber Elaydi, An Introduction to difference equations, $\S 8.2$ Poincare theorem.