# Tagged Questions

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### How many ways can you show that $\displaystyle \int_0^1 \dfrac{\sin(\pi x)}{(1-x)^2}\ \mathrm{d}x\$ is divergent?

How many ways can you show that this integral is divergent? $\displaystyle\int_0^1 \dfrac{\sin(\pi x)}{(1-x)^2} \,dx$ The only way I was able to show this was using a hint which was given to me that ...
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### Investigate the convergence of $\sum a_n$ where $a_n = \int_0^1 \frac{x^n}{1-x}\sin(\pi x) \,dx$

Investigate the convergence of $\sum a_n$ where $a_n = \displaystyle\int_0^1 \dfrac{x^n}{1-x}\sin(\pi x) \,dx$. We have thought about using the dominated convergence theorem to find $\lim a_n$, but ...
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### Which of these compact sets are possible such that they have the following measures?

I am supposed to construct compact sets $K \subset \mathbb{R}$ (if possible) that have the following properties: lambda is the Lebesgue-measure: $\lambda (K^0) = \lambda (K)$. This is easy, just ...
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### $n$th derivative of $(x^2-1)^n$

Define $R_n(x)=\dfrac{d^n}{dx^n}(x^2-1)^n$. Show that $R_n(x)$ is orthogonal to $1,x,\ldots,x^{n-1}$ in $L^2([-1,1])$. Also, what is the value of $R_n(1)$? By definition we have to show that ...