Convergence of a series.

Question

For any $k \in \Bbb N$, if $\{ a_k \} $ satisfies $$ 0 \leqslant a_{k+1} \leqslant \alpha a_k + b_k, \; b_k \geqslant 0, \; 0< \alpha <1, \; \sum_{k=1}^\infty b_k <\infty,$$ then how can I prove $\sum_{k=1}^\infty a_k$ converges?

Answer 1

Hints: Let $B=\sum\limits_{i=1}^{+\infty}b_i$, $A_0=0$ and $A_k=\sum\limits_{i=1}^ka_i$ for every $k\geqslant1$.

(1.) Show that $A_{k+1}\leqslant u(A_k)$ for every $k\geqslant0$, where $u:x\mapsto \alpha x+a_1+B$.

(2.) Show that $A_0\leqslant A_1$.

(3.) Show that $(A_k)_{k\geqslant0}$ is nondecreasing.

(4.) Show that there exists a unique $C$ such that $u(C)=C$.

(5.) Show that $A_k\leqslant C$ for every $k\geqslant0$.

(6.) Conclude.

...And if some steps above are unclear, just yell.

Answer 2

Hint: Consider the Cauchy product of $\sum_{k=1}^\infty b_k$ and $\sum_{k=0}^\infty \alpha^k$.

Answer 3

Consider the series $c_1+c_2+...$ such that $c_1=a_1$ and $c_{k+1}=\alpha c_k + b_k$. Then $0 \le a_k \le c_k$. Now show that $c_1+c_2+...$ converges.