M1 Maths Exponential Functions

With the exponential function expressed in Maclaurin's series, it is:

ex = 1 + x + x2/2! + x3/3! + ... + xn/n! + ...

So the coeff. in (2x + 1)ex is:

(2x + 1)ex = (2x + 1)(1 + x + x2/2! + x3/3! + ... + xn-1/(n - 1)! + xn/n! + ...)

Coeff. of xn:

2/(n - 1)! + 1/n!

= 2n/n! + 1/n!

= (2n + 1)/n!

2011-02-10 15:51:41 補充：
1) 2x 乘以 e^x 入面的 x^(n-1), 得出 2/(n - 1)!

2) 1 乘以 e^x 入面的 x^n, 得出 1/n!

將兩者相加即可

2011-02-10 17:16:04 補充：
2x 乘入去 x^(n-1)/(n-1)! = 2x^n/(n-1)!

1 乘入去 x^n/n! = x^n/(n-1)!

兩個 terms 皆為 x^n 的, 所以加埋就得出 coeff.

同埋2x駛唔駛再乘入去x^(n)/n! ?

不用, 因為會得出 2x^(n + 1)/n!, 不是 x^n 的 term