# Convergence and Integrability of Function Series in Measure Spaces and Applications to Series Expansion Integrals

Prove that $\sum_{n=1}^{\infty } f_{n}$ converges almost everywhere to an integrable function and that

$$\int \sum_{n=1}^{\infty}f_{n} du=\sum_{n=1}^{\infty}\int f_{n}du.$$

2) Use problem 1) to show that$$ \int_{0}^{1}\sin(x)\ln(x)dx=\sum_{n=1}^{\infty} \frac{(-1)^n}{(2n)(2n)!}. $$

Elviegem

40

The answer is accepted.

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