Visions of Mathematics: Series

1. Arithmetic Series

2. Geometric Series

3. Finite Series

$1 + 2 + 3 + .. n = \frac{n(n+1)}{2}$

$2 + 4 + 6 + ..+ 2n = n(n+1)$

$1 + 3 + 5 + ..+ (2n-1) = n^2$ $k + (k+1) + (k+2) + ..+(k+n-1) = \frac{n(2k+n-1)}{2}$

$1^2 + 2^2 + 3^2+.. + n^2 = \frac{n(n+1)(2n+1)}{6}$

$1^3 + 2^3 + 3^3+.. + n^3 = {[\frac{n(n+1)}{2}]} ^2$

$1 + \frac{1}{2} + + \frac{1}{4} + + \frac{1}{8} +... + + \frac{1}{2^n} = 2$

$\frac{1}{1.2} + \frac{1}{2.3} + \frac{1}{3.4} +... + \frac{1}{n(n+1)} = 1$

$1 + \frac{1}{1!} +\frac{1}{2!} + \frac{1}{3!} + .. + \frac{1}{(n-1)!} = e$

Gregory-Leibniz Series (aka Madhava Series)

$\pi = 4 (1-\frac{1}{3} + \frac{1}{5} - \frac{1}{7}+...)$

4. Infinite Series

5. Properties of Convergent Series

Mathematicians are more interesed in convergent series than divergent series.

6. Convergence Tests

7. Alternating Series

8. Power Series

9. Differentation and Integration of Power Series

10. Taylor and Maclaurian Series

Taylor Series

$f(x) = \sum_{n=0}^{\infty} f^{(n)} (a) \frac{(x-a)^n}{n!} = f(a) + f'(a) (x-a) + \frac{f''(a) (x-a)^2}{2!} + .. . + \frac{f^{(n)} (x-a)^n}{n!} + R_n$

Maclaurin Series

$f(x) = \sum_{0}^{\infty} f^{(n)} \frac{x^n}{n!} = f(0) + f^{'}(0)x + \frac{f^{''}(0)x^2}{2!} +..+ \frac{f^{(n)}(0)x^2}{n!} +R_n$

11. Power Series Expansions for Some Functions

$e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!}+ .. + \frac{x^n}{n!}$

$a^x = 1 + x + \frac{x\: lna}{1!} + \frac{{(x\: lna)}^2}{2!}+ \frac{{(x\: lna)}^3}{3!} + .. + \frac{{(x\: lna)}^n}{n!}$

$sin\: x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + .. + \frac{{(-1)}^n x ^{2n+1}}{(2n+1)!} +..$
$cos\: x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + .. + \frac{{(-1)}^n x ^{2n}}{(2n)!} +..$

$tan\: x = x + \frac{x^3}{3} + \frac{2x^5}{15} + \frac{17x^7}{315} + \frac{62x^9}{2835} +..., \left | x \right| < \frac{\pi}{2}$ 1

12. Binomial Series

$(a+b)^n = a^n + \binom{n}{1} a^{n-1}b + \binom{n}{2} a^{n-2}b^2 + ..+ \binom{n}{n-1} ab^{n-1}+ b^n$

$\binom{n}{0} + \binom{n}{1} + ..+ \binom{n}{n} = 2^n$

$\binom{n}{0} - \binom{n}{1} + ..+ (-1)^n \binom{n}{n} = 0$

13. Fourier Series

$f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} (a_n cos \: nx + b_n sin \: nx)$

$a_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x)\, cos \: nx \: dx$

$b_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x)\, sin \: nx \: dx$

The Fourier expansion for a square wave canbe written as

$f(\theta) = \frac{4}{\pi} \: sin\: \theta + \frac{4}{3\pi} \: sin\: 3\theta + \frac{4}{5\pi} \: sin\: 5\theta + ...$

The higher partial sums overshoot near the points of discontinuity 𝜃 = 0, 𝜋, 2𝜋, This is known as Gibbs phenomenon

Unification of Algebra and Geometry - Eulers Formula

$e^{i\theta} = 1 + i\theta - \frac{\theta^2}{2!} - \frac{i\theta^3}{3!} + \frac{i\theta^4}{4!}+..$
$e^{i\theta} = (1 + - \frac{\theta^2}{2!} + \frac{i\theta^4}{4!}+.. ) + i (\theta - \frac{\theta^3}{3!} + \frac{\theta^5}{5!} +.. )$

$e^{i\theta} = cos\: \theta + i\: sin \theta$

Euler derived hundreds of formulas, but this one is often called the most famous formula of all formulas. Feynman called it the amazing jewel.

if θ = 𝜋

$e^{i\pi} = cos(\pi) + isin(\pi) = -1 \\* \\* e^{i\pi} = -1$