Suppose $s$, $t$, and $\delta$ are constants satisfying $0<s<t<1$ and $\delta>1$. An infinite sequence $\{y_k\}_{k=1}^{\infty}$ defined as follows:
The initial term, $y_1$, is the positive root of $(1-s)y_1+sy_1^2=\delta$.
For $k\geq 1$, $y_{k+1}$ is the positive root of $(1-s)y_{k+1}+sy_{k+1}^2=(1+t)y_k-ty_k^2$ given $y_k$.
I have trouble in calculating $\sum_{k=1}^{\infty}(y_k-1)$ although I can show that $\lim_{k\rightarrow\infty}y_k=1$.
Thanks in advance.