We have the inequality
$$\alpha_n(t) \le 4\pi(3\pi)^{1/3} \exp\left\{\int_0^t(1+3F_P(\sigma)) \, d\sigma \right\} \cdot \int_0^t P(\sigma)^2(3CD_1^2)^{1/3}\alpha_{n-1}(\sigma) \, d\sigma$$
for $n=2,3,\ldots$. (We notice that $\alpha_n$ appears on both sides of the inequality.) 

Why does it follow that the infinite series
$$\sum_{n=1}^\infty \alpha_n(t)$$
converges locally uniformly on $\mathbb R_0^+$ (that is, converges on $[0,T]$ for $0 \le t \le T$)?

This comes from page 354 of the journal that contains the paper "**[Global symmetric solutions of the initial value problem of stellar dynamics][1]**" by Jurgen Batt.


  [1]: http://www.sciencedirect.com/science/article/pii/0022039677900493