A collection of some of my notes. Some of them were written for teaching purposes, while others were written for entertainment. In case you find any blunders/errors/typos/grammatical mistakes (however small or big it might be), I would greatly appreciate if you could let me know.

#### Algebra

- Eisenstein’s criterion for polynomial irreducibility
- Vandermonde matrix
- AM-GM inequality
- Newton’s and Maclaurin’s inequality
- Cauchy Schwarz Inequality
- Diagonalization of symmetric matrices
- Eigenvalues and eigenvectors of circulant matrices
- Least square and least norm
- Gershgorin Circle theorem
- Young’s inequality

#### Analysis

- Different ways of proving \(\displaystyle\int_{\mathbb{R}} e^{-x^2}dx\) _
- Evaluating \(\displaystyle \int_0^{\pi/2} \sin^k(t) dt\) or \(\displaystyle \int_0^{\pi/2} \cos^k(t) dt\)
- Wallis formula: \(\dfrac{\pi}2 = \displaystyle \prod_{n=1}^{\infty} \left(\dfrac{2n}{2n-1} \cdot \dfrac{2n}{2n+1}\right)\) and asymptotic of the central binomial coefficient _
- De Moivre's and Stirling's formula
- An identity on product of sines
- Ostrowski's Theorem
- Fresnel integral
- \(\displaystyle \int_0^{\pi}\ln(\sin(x))dx = -\pi \ln(2)\)
- Frullani Integral
- Bernoulli Inequality
- Summation of sine and cosine
- Evaluating \(\displaystyle \int_0^{\pi/2} \dfrac{dt}{1+r\sin(t)}\) and related integrals
- An elegant way to prove that \(\lim_{n \to \infty} n^{1/n}=1\)
- An infinite product identity; Solution to SIAM problem posted by Carlo Sanna
- Divergence of harmonic series
- Completeness of \(\mathbb{R}\)
- A new way of regularizing sums of the form \(1^k+2^k+\cdots+n^k+\cdots\)
- Motivation for doing real analysis course
- Weierstrass approximation theorem
- Markov and Chebyshev inequalities
- Holder's inequality
- Minkowski's inequality
- Cantor's intersection theorem
- Bolzano Weirstrass Theorem