`f(x)=1/(1+x)^4` Use the binomial series to find the Maclaurin series for the function.

Expert Answers

An illustration of the letter 'A' in a speech bubbles

Recall binomial series  that is convergent when `|x|lt1` follows: 

`(1+x)^k=sum_(n=0)^oo _(k(k-1)(k-2)...(k-n+1))/(n!)`

or`(1+x)^k= 1 + kx + (k(k-1))/(2!) x^2 + (k(k-1)(k-2))/(3!)x^3 +(k(k-1)(k-2)(k-3))/(4!)x^4-` ...

 For the given function `f(x) =1/(1+x)^4` , we may  apply Law of Exponents: `1/x^n = x^(-n)` to rewrite it as:

`f(x) = (1+x)^(-4)`

This now resembles `(1+x)^k` for binomial series.  

By comparing "`(1+x)^k` " with "`(1+x)^(-4)` ", we have the corresponding values:

`x=x` and `k = -4` .

 Plug-in the values  on the formula for binomial series, we get:

`(1+x)^(-4)=sum_(n=0)^oo ((-4)(-4-1)(-4-2)...(-4-n+1))/(n!)x^n`

               `= 1 + (-4)x + ((-4)(-4-1))/(2!) x^2 + ((-4)(-4-1)(-4-2))/(3!)x^3 +((-4)(-4-1)(-4-2)(-4-3))/(4!) x^4-` ...

` = 1 + (-4)x + ((-4)(-5))/(2!) x^2 + ((-4)(-5)(-6))/(3!)x^3 +((-4)(-5)(-6)(-7))/(4!) x^4-` ...

` = 1 -4x + 20/(2!) x^2 -120/(3!)x^3 +840/(4!)x^4-` ...

` = 1- 4x +10x^2 -20x^3 +35x^4-` ...

Therefore, the Maclaurin series  for  the function `f(x) =1/(1+x)^4` can be expressed as:

`1/(1+x)^4 =1- 4x +10x^2 -20x^3 +35x^4-` ...

See eNotes Ad-Free

Start your 48-hour free trial to get access to more than 30,000 additional guides and more than 350,000 Homework Help questions answered by our experts.

Get 48 Hours Free Access
Approved by eNotes Editorial