# 自然對數e

[https://zh.wikipedia.org/wiki/E\_(数学常数\\](https://zh.wikipedia.org/wiki/E_%28数学常数\\)

往下拉到定義部分

## 1.

```
定義 e 爲下列極限值：
```

![](/files/-M0u4ZYoQiMX9aRSfscZ)

可用程式計算

> n趨近極大值

```javascript
function a (n){return (1+1/n) ** n }

// a(9000000000)
// 2.718282053219686
```

以及

![](/files/-M0u4ZYq8CQ6roGxYB8z)

> n趨近極小值

```javascript
function b (n){return (1+n) ** (1/n) }

// b(0.000000000001)
// 2.7185234960372378
```

> 但javascript有最大限制
>
> <http://stackoverflow.com/questions/43729790/natural-logarithm-e-1>

## 2.

```
定義 e 爲下列無窮級數之和：
```

可用程式計算

```
//fact用來計算階層的倒數
function fact(x) {
   if(x==0) {
      return 1;
   }
   return 1 / x * fact(x-1) ;
}


function c(n) {
  var total = fact(n);
  for(i = 1; i <= n; i++ ){
    total += fact(n - i)
  }
 return total;
}

//c(345)
```

\>以上兩者給n到一數以上時都會為2.7.......

## 當![{\displaystyle x=e}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8ba47ab1931fc4886c5da08831962cc141d20655)時函數![f(x) = \sqrt\[x\]{x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e15600c8e263f973cca6db5347531ea5a6846aa8)有最大值

```javascript
Math.newsqrt = function(a, b) {
  if(b > 0) {
    a = Math.sqrt(a);
    return Math.newsqrt(a, b - 1);
  } else {
    return a
  }
}

Math.newsqrt(Math.exp(1), Math.exp(1))
```

<http://blog.udn.com/cchahacaptain/4565752>


---

# Agent Instructions: Querying This Documentation

If you need additional information that is not directly available in this page, you can query the documentation dynamically by asking a question.

Perform an HTTP GET request on the current page URL with the `ask` query parameter:

```
GET https://easonwang.gitbook.io/trigonometric-functions/zi-ran-dui-shu-e.md?ask=<question>
```

The question should be specific, self-contained, and written in natural language.
The response will contain a direct answer to the question and relevant excerpts and sources from the documentation.

Use this mechanism when the answer is not explicitly present in the current page, you need clarification or additional context, or you want to retrieve related documentation sections.