16 Proving an identity relating to the complex modulus: $z\bar{a}+\bar{z}a \leq 2|a||z|$ 7 How to prove that $\int_{-\infty}^{\infty}f'(t)\exp(-t^2)\text{d}t$ is finite when $\lim_{|t|\to \infty}f(t)\exp(-t^2) = 0$? 7 Why does $\ker(T) = \{0\} \Leftrightarrow T$ is injective 6 Induction differential proof 6 Find $f(1)+\cdots+f(60)$

### Reputation (4,367)

 +10 Why does $\ker(T) = \{0\} \Leftrightarrow T$ is injective +10 Why $\int_0^\infty \sin^2(x)/x^2 dx$ converges, and $\int_0^\infty \cos^2(x)/x^2 dx$ does not? +10 Proof for bracketing a positive series +10 Harmonic series and the binomial theorem

### Questions (5)

 9 Show that polynomial $X^{n+1}-aX^n+aX-1$ has only roots of module $1$ 5 Expected value of the shifted inverse of a binomial random variable, and application 1 Explaining a probability for a repeated drawing without replacement 1 Property of triangles with integer coordinates. 1 Equivalent of the integral of a function sequence

### Tags (162)

 60 real-analysis × 30 25 inequality × 6 38 calculus × 24 22 complex-numbers × 7 38 sequences-and-series × 24 19 probability × 23 30 algebra-precalculus × 15 18 linear-algebra × 12 25 limits × 11 17 integration × 9

### Bookmarks (28)

 40 In a village, $90\%$ of people drink Tea, $80\%$ Coffee, $70\%$ Whiskey, $60\%$ Gin. Nobody drinks all four. What percentage of people drinks alcohol? 28 Structures emerging in a discrete plot of palindromic numbers 18 Is there a bijection $f: \mathbb{N} \rightarrow \mathbb{N}$ such that the series $\sum\limits_n \frac{1}{n+f(n)}$ is convergent? 13 How to prove that $\sum_{k=1}^n\frac{1}{\sqrt[n]{k!} }\sim \frac{n}{\ln n}$ 10 Properties of the Minimum of Two Poisson Random Variables