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](http://www.sciencedirect.com/science/article/pii/0022039677900493)**" by Jurgen Batt.



Inequality implies locally uniform convergence of a series
Some estimates like $\alpha\_n(s)\leqslant A(T)^n/n!$ for all $s\in [0,T]$ should hold by induction, if we suppose something moderate on your functions, which depend on $\sigma$. This yields desired uniform convergence.


