### Questions (148)

 8 How do I compare $\sqrt{2}$ and $\pi^{1/ \pi}$? 6 Let $(X,\mathscr T)$ be a metrizable space such that every metric that generates $\mathscr T$ is bounded. Prove that $X$ is compact. [duplicate] 6 $\sum\limits_{n=1}^\infty \log(1+a_n)$ converges absolutely $\iff\sum\limits_{n=1}^\infty a_n$ converges absolutely. 6 How to derive the formula for $\sum_{n=0}^{\infty}\tan^{-1}(\frac{x^{2}}{n^{2}})$ [duplicate] 5 If $|z| < 1$, prove that $\Re \left(\frac{1}{1 - z} \right) > \frac{1}{2}$.

### Reputation (2,694)

 +5 Singularity of the function $f(z)=\sin\left(\cos\left(\frac{1}{z}\right)\right)$ at the point $z=0$ is … +5 Let $f(z)=z+\frac{1}{z}$ for $z\in \mathbb C$ with $z\ne 0$. Which of the following is always true? +5 $\sum_{n=1}^\infty a_n$ converges and $\sum_{n=1}^\infty|a_n|$ diverges. Then Radius of convergence? +5 Errors and doubts in the proof.

### Answers (97)

 4 Union and intersection of functions 3 Show that $f(x)=\alpha x$ for some $\alpha \in R$ 3 Let f : [0, 1] → [0,1]. Suppose that f attains every value between 0 and 1 in the interval [0, 1], then f is continuous in the interval [0, 1]. 3 To find $\dim(W_1+W_2)$ 2 Limit Inferior of Sequence $x_n = 2^n$

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