Convergence of series $a(n)$, where $a(n+1) = p + qa(n)$, $n \geq 1$

Question

Suppose for $n \geq 1$, we have for some constants $p,q$

$a(n+1) = p + qa(n)$.

Conditions on $p$ and $q$ for which the series converges?

So, we have the series as

$a(1)+a(2)+a(3)+\ldots $

$a(1) + [p + qa(1)] + [p + pq + q^2a(1)] + [p+pq+pq^2 + q^3a(1)] + \ldots p + pq + pq^2 + \ldots + pq^n + q^na(1) $

I was thinking that sequence of partial sums should converge for the series to converge.

$S_{n} = a(1)+a(2)+a(3)+\ldots+a(n)$ $S_{n} = (n-1)p + a(1) [q+q^2 +\ldots+q^n] + (n-2)pq + (n-3)pq^2 + \ldots$

but got stuck now.

Answer 1

Note first that convergence requires $\lim_{n\to\infty}a(n)=0$, so $0=p+q0$ i.e. $p=0$. Then $a(n)=a(1)q^{n-1}$, whence $|q|<1$. This is sufficient too, as we get a convergent geometric series.