Convergence and Integrability of Function Series in Measure Spaces and Applications to Series Expansion Integrals
1) Let $(X,\Sigma,\mu)$ be a measure space and for each n suppose that $f_{n}:X\rightarrow \mathbb{R} $ is an integrable function such that $\sum_{n=1}^{\infty } \int |f_{n}|du$ is convergent.
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
$$ \int_{0}^{1}\sin(x)\ln(x)dx=\sum_{n=1}^{\infty} \frac{(-1)^n}{(2n)(2n)!}. $$
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
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Poincare
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