# Series (Calculus)

[►Jump back to previous note: Series (High school level)](https://github.com/solomonxie/solomonxie.github.io/issues/44#issuecomment-375569163)

Explicit Sequence vs. Recursive Sequence:

> Explicit sequence would be presented as: `a𝓃 = a₁ · kⁿ⁻¹`. Recursive sequence would be presented as: `a₁ = 3, a𝓃 = k · a𝓃₋₁`

Sequence vs. Series:

> Sequence is a ***LIST*** of numbers, Series is ***a NUMBER***: the SUM of a sequence.

Convergence vs. Divergence:

> Convergence means the **limit** of a function **EXISTS**. Divergence means the limit **DOES NOT EXISTS**.

## `Geometric Series in 𝚺 Notation`

[▼Refer to Cool Math: Geometric Series](http://www.coolmath.com/algebra/19-sequences-series/08-geometic-series-02) ![image](https://user-images.githubusercontent.com/14041622/41806975-05a66fac-76fa-11e8-80c4-dd9cacb3dc10.png)

### Example

![image](https://user-images.githubusercontent.com/14041622/41806978-167f350c-76fa-11e8-91a3-4ce9d734ec15.png)

## `Infinite Sequence (convergence | divergence)`

[►Jump to practice: Sequence convergence/divergence](https://www.khanacademy.org/math/ap-calculus-bc/bc-series/modal/e/convergence-and-divergence-of-sequences)

### Example

![image](https://user-images.githubusercontent.com/14041622/41806326-79e751d0-76ee-11e8-8daf-5305147d0c65.png)

* Easiest way: Apply the \`L'hopital's Rule, take both Top's & Bottom's derivatives until both of them become numbers.
* So we get: `1/3`.

## `Finite Geometric Series`

[►Jump to practice: Finite geometric series](https://www.khanacademy.org/math/ap-calculus-bc/bc-series/modal/e/geometric-series--1)

### Example

![image](https://user-images.githubusercontent.com/14041622/41806346-fffbf8fc-76ee-11e8-96f7-f9f05cbd081d.png) Solve:

* By using the `Geometric Series formula`, we get the informations as below:
  * **Common ratio**: `r = -2`
  * **Amount of items**: `n = 20`. Because `k` starts from 0, so there're 20 terms.
  * **Initial term**: `a₀ = -4`
* We calculate and get the result as below:

  ![image](https://user-images.githubusercontent.com/14041622/41806402-f872cd58-76ef-11e8-8d02-297df7f4f81d.png)

## `Partial Sums`

`Partial sums` is just a fancy word for `Finite series`, because it's a a **part** of infinite series.

### Example

![image](https://user-images.githubusercontent.com/14041622/41806657-ab1c858a-76f4-11e8-9f6c-dabd7dfbced4.png) Solve:

* The tricky part is **how to count the amount of terms**.
* Since `n` starts from 1, so there're 11 terms, which means we're to calculate `S₁₁`.
* `S₁₁ = 88/16 = 11/2`

### Example

![image](https://user-images.githubusercontent.com/14041622/41806687-839d74e6-76f5-11e8-8f60-bcde99c82811.png) Solve:

* The tricky here is that: `a𝓃 = S𝓃 - S𝓃-1`, because `S𝓃 = a₁ + a₂ + a₃ +.... + a𝓃-1 + a𝓃`.
* So the result is:

  ![image](https://user-images.githubusercontent.com/14041622/41806716-de76dede-76f5-11e8-9bdc-b6929d0a45f4.png)