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# Series Cheat Sheet

### Arithmetic Series

$${s_n = n \frac{a_1 + a_n}{2}}$$

$${s_n = n \frac{2a_1 + d (n-1)}{2}}$$

$${a_n = a_1 + (n -1)d}$$

### Geometric Series

$${\sum_{n=0}^{\infty} a (r^n) = \frac{a}{1 -r}, |r| < 1}$$

$${\sum_{n=1}^{\infty} a (r^{n-1}) = \frac{a}{1 -r}, |r| < 1}$$

### Divergence Test

If $${\lim_{n \to \infty} a_n \ne 0}$$ then $${\sum a_n}$$ diverges

### Comparison Test

For $${\sum a_n}$$ and $${\sum b_n}$$ where $${a_n \le b_n}$$ for all n.

and $${a_n > 0}$$ and $${b_n > 0}$$ then

If $${\sum b_n}$$ is convergent then $${\sum a_n}$$ is convergent.

If $${\sum a_n}$$ is divergent then $${\sum b_n}$$ is divergent.

### P-series Test

For $${\sum_\limits{n=0}^{\infty} \frac{1}{n^p}}$$

If $${p \gt 1}$$ then the series converges

If $${p \le 1}$$ then the series diverges

### Integral Test

For $${\sum a_n}$$ if $${a_n}$$ is continous, positive and decreasing then

If $${\int_{k}^{\infty}{a_n \, dx} }$$ is convergent then $${\sum a_n}$$ is convergent.

If $${\int_{k}^{\infty}{a_n \, dx} }$$ is divergent then $${\sum a_n}$$ is divergent.

### Limit Comparison Test

For $${\sum a_n}$$ and $${\sum b_n}$$

if $${L = \lim\limits_{n \to \infty} \frac{a_n}{b_n}}$$ where $${0 < L < \infty}$$

then either both $${\sum a_n}$$ and $${\sum b_n}$$ are convergent or

both series are divergent.

### Alternating Series Test

For $${\sum a_n}$$ where $${a_n = (-1)^n b_n, b_n > 0}$$

Then if $${L = \lim\limits_{n \to \infty} {b_n = 0}}$$ and $${b_n}$$ is decreasing

then $${\sum a_n}$$ is convergent.

### Ratio Test

For $${\sum a_n}$$ and $${L = \lim\limits_{n \to \infty} {|\frac{a_{n+1}}{a_n}}|}$$

If $${L < 1}$$ then $${\sum a_n}$$ is absolutely convergent (hence convergent).

If $${L > 1}$$ then $${\sum a_n}$$ is divergent.

If $${L = 1}$$ then the test is inconclusive.

### Root Test

For $${\sum a_n}$$ and $${L = \lim\limits_{n \to \infty} {\sqrt[n]{|a_n|}}}$$

If $${L < 1}$$ then $${\sum a_n}$$ is absolutely convergent (hence convergent).

If $${L > 1}$$ then $${\sum a_n}$$ is divergent.

If $${L = 1}$$ then the test is inconclusive.

### Absolute Convergence

If $${\sum |a_n|}$$ is convergent then $${\sum a_n}$$ is absolutely convergent.

If a series is absolutely convergent then it is also convergent.

If $${\sum |a_n|}$$ is divergent and $${\sum a_n}$$ is convergent then the series is conditionally convergent.

### Telescoping Series

Series whose terms will cancel to a finite number of terms

Example:

$${\sum_{n=1}^{\infty} \frac{1}{n(n+1)} = \sum_{n=1}^{\infty} (\frac{1}{n} - \frac{1}{n+1}) }$$

$${\require{cancel}= (1 - \cancel{\frac{1}{2}}) + (\cancel{\frac{1}{2}} - \cancel{\frac{1}{3}}) + (\cancel{\frac{1}{3}} - \cancel{\frac{1}{4}}) + (\cancel{\frac{1}{4}} - \cancel{\frac{1}{5}}) + \dots = 1}$$

### Harmonic Series

Divergent infinite series of the form $${\sum \frac{1}{an+b}}$$

Example:

$${\sum_{n=1}^{\infty} \frac{1}{n}}$$

### Power Series

$${\sum_{n=0}^{\infty} C_n (x - a)^n}$$

radius of convergence: $${|x - a| < R}$$

interval of convergence: $${a - R < x < a + R}$$ but check the endpoints for equality.

$${R = \lim\limits_{n \to \infty} |\frac{C_n}{C_{n + 1}}|}$$

### Power Series - Geometric Series

$${\sum_{n=0}^{\infty} x^n = \frac{1}{1 - x}, |x| < 1, R = 1}$$

### Taylor Series

$${\sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!} (x - a)^n}$$

### MacLaurin Series

$${\sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!} (x)^n}$$

### Some common power series:

$${e^x = \sum_{n=0}^{\infty} \frac{(x)^n}{n!} = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!}}$$

$${cos(x) = \sum_{n=0}^{\infty} \frac{(-1)^n(x)^{2n}}{(2n)!} = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!}}$$

$${sin(x) = \sum_{n=0}^{\infty} \frac{(-1)^n(x)^{2n+1}}{(2n+1)!} = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!}}$$

$${cosh(x) = \sum_{n=0}^{\infty} \frac{(x)^{2n}}{(2n)!} = 1 + \frac{x^2}{2!} + \frac{x^4}{4!} + \frac{x^6}{6!}}$$

$${sinh(x) = \sum_{n=0}^{\infty} \frac{(x)^{2n+1}}{(2n+1)!} = x + \frac{x^3}{3!} + \frac{x^5}{5!} + \frac{x^7}{7!}}$$

#### Useful facts (credit blackpenredpen):

For $${n \to \infty, p > 0, b > 1}$$

$${ln(n) < \sqrt{n} < n^p < b^n < n! < n^n}$$

#### Definition of e:

$${\lim\limits_{n \to \infty} (1 + \frac{a}{n})^{bn} = e^{ab}}$$

#### Don't use L'Hospitals Rule!:

$${\lim\limits_{\theta \to 0} \frac{sin(\theta)}{\theta} = 1}$$