# Integral Test

[`►Jump over to have practice at Khan academy: Integral test.`](https://www.khanacademy.org/math/ap-calculus-bc/bc-series/modal/e/integral-test) [Refer to article from tkiryl: The Integral Test](http://www.tkiryl.com/Calculus/Section_8.3--The_Integral_and_Comparison_Tests/The_Integral_and_Comparison_Tests.html) [Refer to Khan academy: Integral Test](https://www.khanacademy.org/math/ap-calculus-bc/bc-series/modal/v/integral-test-intuition)

[▼Refer to awesome article from xaktly: Integral Test](http://xaktly.com/IntegralTest.html)

![image](https://user-images.githubusercontent.com/14041622/41808008-1f857dc2-770a-11e8-903e-0db037158421.png)

![image](https://user-images.githubusercontent.com/14041622/41899074-19a5e2ce-795e-11e8-8b1a-1f20c7d4748c.png)

## Conditions of Integral test

Assume the series `a𝖓` can be represented as a function `f(x)`. There are a few limitations for it to use the Integral test:

* `f(x)` MUST BE continuous.
* `f(x) > 0`. It MUST BE a **positive** function.
* `f'(x) < 0`. It's MUST BE **decreasing**.

## Using Integral test

![image](https://user-images.githubusercontent.com/14041622/41809149-128605fe-771b-11e8-8627-cc64ed8df4ab.png)

### Example

![image](https://user-images.githubusercontent.com/14041622/41809122-78b3cb96-771a-11e8-887a-8347e04b0fc1.png) Solve: ![image](https://user-images.githubusercontent.com/14041622/41809128-a32d5932-771a-11e8-8b97-699ab6287861.png)

## Understanding Integral test

The `Integral test` has introduced the idea of calculating the total area under the function:

* The series has step of 1, which means `Δx = 1`
*   We can sum the areas (which equals the series itself):&#x20;

    ![image](https://user-images.githubusercontent.com/14041622/41808077-d6a26876-770a-11e8-9ef7-18558302e24f.png)
*   But when we are to **INTEGRATE** the function area under the function:

    ![image](https://user-images.githubusercontent.com/14041622/41808108-5fd348ea-770b-11e8-9a54-09729ac28dc1.png)
* The `dx` is infinitely small rather than a fixed number `Δx = 1`.
* As result, the **INTEGRAL** is almost always greater than the **SERIES AREAS**.

As been said above, we got this conclusion: ![image](https://user-images.githubusercontent.com/14041622/41807976-ae338b00-7709-11e8-8ab9-5cdde0e5f4b3.png)

> **Notice: DO NOT use the** `Integral Test` **to EVALUATE series, because in general they are NOT equal.**
