# Suppose $k_n\subset[0,\infty)$ with $k_n\to1$ as $n\to\infty$ and $\alpha_n$ is in $[0,1)$ is $\sum\alpha_n(k_n(M+1)-1)<\infty$ where $M>1$

we am working on a mapp to show that the Mann iteration converges. At the end of the computation we came up the series $$\sum\alpha_n(k_n(M+1)-1)$$, from previous work it is known that the series $$\sum\alpha_n(k_n-1)<\infty$$.

we wish to know if $$\sum\alpha_n(k_n(M+1)-1)<\infty$$ and if not why?