# 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

41

## Answer

**Answers can only be viewed under the following conditions:**

- The questioner was satisfied with and accepted the answer, or
- The answer was evaluated as being 100% correct by the judge.

1 Attachment

Poincare

133

The answer is accepted.

Join Matchmaticians Affiliate Marketing
Program to earn up to a 50% commission on every question that your affiliated users ask or answer.

- answered
- 279 views
- $20.00

### Related Questions

- real analysis
- Need help with integrals (Urgent!)
- Existence of a Divergent Subsequence to Infinity in Unbounded Sequences
- Derivative of $\int_{\sin x}^{x^2} \cos (t)dt$
- What is the Lebesgue density of $A$ and $B$ which answers a previous question?
- Prove that $A - B=A\cap B^c$
- do not answer
- Riemann Sums for computing $\int_0^3 x^3 dx$