Convergence of Partial Sums

Question

Suppose we have the series of functions $$\DeclareMathOperator{\Dm}{d\!} F(x)=\sum_{n=1}^{\infty} f_n(x) $$ where convergence is uniform.
Additionally, consider the partial functions of the series $$ F_m (x)=\sum_{n=1}^{m} f_n(x) $$ satisfying the condition $$ \left\|\int_{0}^{1} \frac{F_m(x)}{x^{3}}\Dm x\right\|<C $$ for some positive constant $C$, for every $m$.
If it is also true that $$ \int_{0}^{1} \frac{\sum_{n=m}^{\infty} f_n(x)}{x^{3}} \to 0 $$ as $m \to \infty$, can we then claim that $$ \int_{0}^{1} \frac{F_m(x)}{x^{3}}\Dm x \to \int_{0}^{1} \frac{F(x)}{x^{3}}\Dm x $$ as $m \to \infty$?

Answer 1

You have : $$ \int_0^1\frac{F(x)}{x^3}dx-\int_0^1\frac{F_m(x)}{x^3}dx=\int_0^1\frac{F(x)-F_m(x)}{x^3}dx\underset{m\rightarrow+\infty}{\longrightarrow}0 $$ by hypothesis. You don't need the upper bound assumption nor the uniform convergence, but you do need that your integrals exist, there may be a problem at $0$.