crskhr

 18 How can I show that $\begin{pmatrix} 1 & 1 \\ 0 & 1\end{pmatrix}^n = \begin{pmatrix} 1 & n \\ 0 & 1\end{pmatrix}$? 12 Prove: $\sin (\tan x) \geq {x}$ 11 Find the value of: $\sum_{n=1}^{50}n(n!)$ 10 Is $\left( {{2}^{x}}-1 \right)\left( {{5}^{x}}-1 \right)$ a square number for integer $x>1$ 9 Alternating sum of binomial coefficients multiplied by (1/k+1)

### Reputation (3,990)

 +65 Showing that $\lim_{n \rightarrow \infty}\left(1+\frac{1}{n}+\frac{1}{n^2}\right)^n = e$ +25 The number of quadratic residues modulo p in the set ${1,2,…,p-1}$ +10 How can I show that $\begin{pmatrix} 1 & 1 \\ 0 & 1\end{pmatrix}^n = \begin{pmatrix} 1 & n \\ 0 & 1\end{pmatrix}$? +5 Counting the number of inversions for the function $f(x+yp)=qx+y$

### Questions (17)

 14 How does one evaluate $\int \frac{\sin(x)}{\sin(5x)} \ dx$ 12 If $f(1)=1$, then is it true that $f(n)=n$ for all $n \in \mathbb{N}\cup\{0\}$. 10 How to show $\sum_{k=0}^{n}\binom{n+k}{k}\frac{1}{2^k}=2^{n}$ [duplicate] 7 Relatively prime elements forming an arithmetic progression 7 $\sum_{n=1}^{\infty}\frac{1}{n^2} <\frac{33}{20}$ using elementary inequalities

### Tags (77)

 56 calculus × 22 29 limits × 8 49 sequences-and-series × 16 24 summation × 4 45 integration × 17 21 number-theory × 12 35 elementary-number-theory × 21 18 matrices 35 trigonometry × 13 18 linear-algebra

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