# Maclaurin Series of Common functions

[►Refer to Wiki: List of Maclaurin series of some common functions](https://en.wikipedia.org/wiki/Taylor_series#List_of_Maclaurin_series_of_some_common_functions)

Maclaurin series of these common functions are very useful, which we really want to **memorize**.

| Function           | Maclaurin Series                                                                                               |
| ------------------ | -------------------------------------------------------------------------------------------------------------- |
| sin(x)             | ![image](https://user-images.githubusercontent.com/14041622/42365046-f3794fbc-812f-11e8-84c0-c6e1aecf1b66.png) |
| cos(x)             | ![image](https://user-images.githubusercontent.com/14041622/42365056-fbf4c694-812f-11e8-89c6-8cb191427016.png) |
| tan(x)             | ![image](https://user-images.githubusercontent.com/14041622/42365075-06c91cb4-8130-11e8-89f8-2e93d7c6e4df.png) |
| sec(x)             | ![image](https://user-images.githubusercontent.com/14041622/42365080-0d658760-8130-11e8-876e-b1ba6348b4fa.png) |
| eˣ                 | ![image](https://user-images.githubusercontent.com/14041622/42364980-a87f8224-812f-11e8-8e95-81d2750cb3cf.png) |
| Geometric series 1 | ![image](https://user-images.githubusercontent.com/14041622/42365162-450be27c-8130-11e8-81f7-cdcf2a0931a3.png) |
| Geometric series 2 | ![image](https://user-images.githubusercontent.com/14041622/42365179-5066964e-8130-11e8-8fa8-73c8da4cf954.png) |
| Geometric series 3 | ![image](https://user-images.githubusercontent.com/14041622/42365183-54348ccc-8130-11e8-8975-c9f6fa211a32.png) |

## Example

![image](https://user-images.githubusercontent.com/14041622/42364580-2e227938-812e-11e8-89f4-de4979f77b2a.png) Solve:

*   This Maclaurin series has a clear pattern of `cos(x) Maclaurin series`:

    ![image](https://user-images.githubusercontent.com/14041622/42364638-672415a2-812e-11e8-8b30-0b2a6fa4e1bb.png)
*   In this case, it's:

    ![image](https://user-images.githubusercontent.com/14041622/42364653-7e942baa-812e-11e8-93c8-a4d17062a59b.png)
*   So we could convert the series back to the trig function:

    ![image](https://user-images.githubusercontent.com/14041622/42364676-8f36be1e-812e-11e8-96d3-3d0821335888.png)

## Example

![image](https://user-images.githubusercontent.com/14041622/42365320-bb3528e6-8130-11e8-8ee6-ee6a96934cbf.png) Solve: ![image](https://user-images.githubusercontent.com/14041622/42365343-c6d55f7c-8130-11e8-9bc7-01dd5e40dd34.png)

## Example

![image](https://user-images.githubusercontent.com/14041622/42505266-da5133b2-8470-11e8-92f8-09468c2c069b.png) Solve:

* Looks quite a complicated series, but we look at the `factorial dominators`, we found that resembles the pattern of the `sin(x)`'s maclaurin series.
* By comparing to the `sin(x) maclaurin series`, we notice that `x = π/2`, and all the rest are the same.
*   So rewrite the series to the function:

    ![image](https://user-images.githubusercontent.com/14041622/42505445-564793e4-8471-11e8-89f0-1e317af23d74.png)
* And we get the result is `1`.

## Example

![image](https://user-images.githubusercontent.com/14041622/42506152-9cf2a0ca-8473-11e8-8fdd-f5257514ad6e.png) Solve:
