Suppose we have the series of functions:

\begin{equation} F(x)=\sum_{n=1}^{\infty} f_n(x) \end{equation}

where convergence is uniform.

Additionally, consider the partial functions of the series:

\begin{equation} F_m (x)=\sum_{n=1}^{m} f_n(x) \end{equation}

Suppose that:

\begin{equation} \int_{0}^{1} \frac{F_m(x)}{x^{3}} ,dx \end{equation}

forms a Cauchy sequence, hence convergent.

Can we then claim that:

\begin{equation} \int_{0}^{1} \frac{F_m(x)}{x^{3}} ,dx \to \int_{0}^{1} \frac{F(x)}{x^{3}} ,dx \end{equation}

as $m \to \infty$?

Obs: \begin{equation} \int_{0}^{1} \frac{F(x)}{x^{3}} ,dx \end{equation}

exist

Suppose we have the series of functions:

\begin{equation} F(x)=\sum_{n=1}^{\infty} f_n(x) \end{equation}

where convergence is uniform.

Additionally, consider the partial functions of the series:

\begin{equation} F_m (x)=\sum_{n=1}^{m} f_n(x) \end{equation}

Suppose that:

\begin{equation} \int_{0}^{1} \frac{F_m(x)}{x^{3}} ,dx \end{equation}

forms a Cauchy sequence, hence convergent.

Can we then claim that:

\begin{equation} \int_{0}^{1} \frac{F_m(x)}{x^{3}} ,dx \to \int_{0}^{1} \frac{F(x)}{x^{3}} ,dx \end{equation}

as $m \to \infty$?

observation: \begin{equation} \int_{0}^{1} \frac{F(x)}{x^{3}} ,dx \end{equation}

exist

Suppose we have the series of functions:

\begin{equation} F(x)=\sum_{n=1}^{\infty} f_n(x) \end{equation}

where convergence is uniform.

Additionally, consider the partial functions of the series:

\begin{equation} F_m (x)=\sum_{n=1}^{m} f_n(x) \end{equation}

Suppose that:

\begin{equation} \int_{0}^{1} \frac{F_m(x)}{x^{3}} ,dx \end{equation}

forms a Cauchy sequence, hence convergent.

Can we then claim that:

\begin{equation} \int_{0}^{1} \frac{F_m(x)}{x^{3}} ,dx \to \int_{0}^{1} \frac{F(x)}{x^{3}} ,dx \end{equation}

as $m \to \infty$?

Suppose we have the series of functions:

\begin{equation} F(x)=\sum_{n=1}^{\infty} f_n(x) \end{equation}

where convergence is uniform.

Additionally, consider the partial functions of the series:

\begin{equation} F_m (x)=\sum_{n=1}^{m} f_n(x) \end{equation}

Suppose that:

\begin{equation} \int_{0}^{1} \frac{F_m(x)}{x^{3}} ,dx \end{equation}

forms a Cauchy sequence, hence convergent.

Can we then claim that:

\begin{equation} \int_{0}^{1} \frac{F_m(x)}{x^{3}} ,dx \to \int_{0}^{1} \frac{F(x)}{x^{3}} ,dx \end{equation}

as $m \to \infty$?

Obs: \begin{equation} \int_{0}^{1} \frac{F(x)}{x^{3}} ,dx \end{equation}

exist