Verify that a sequence is a recursive sequence

Question

I'm trying to understand the nature of the following expression: $\begin{equation} \label{eq:1} s_n=s_{n-1}+x_n \end{equation}$ where $$x_n=\dfrac{1}{2}m_n^2\left(\sum_{k=1}^N \dfrac{1}{4\lambda_{k,n}}\right)^{0.5}+2m_m\sum_{i=n-1}^1m_i\left(\sum_{k=1}^N\dfrac{1}{4(\lambda_{k,n}+\lambda_{k,i})}\right)^{0.25}$$ and $$x_1= \dfrac{1}{2}m_1^2\left(\sum_{k=1}^N \dfrac{1}{4\lambda_{k,1}}\right)^{0.5}.$$Both $\lambda$ and $m$ are positive scalars. My attempt to characterize the convergence of this expression was considering $s_{n+1}=s_{n}+x_{n+1}$ as a recursive sequence of the form $s_{n+1}=f(s_n)$ and so solve the relation $l=f(l)$, where $l= \lim_{n\rightarrow+\infty}s_{n}$. But now I wonder is $s_{n+1}=f(s_n)$ a recursive sequence?