Question

Find the Maclaurin polynomials of orders n=0,1,2,3 and 4 and then find the nth Maclaurin polynomials for the function in sigma notation. Enter the Maclaurin polynomials below for 2/(1+x). p0(x), p(1)x, p(2)x, p(3)x, p(4)x, pn(x)=sum n m=0

Answer 1

The Maclaurin polynomials for the function 2/(1+x) of orders 0 through 4 are successively

`p0(x) = 2, p1(x) = 2 - 2x, p2(x) = 2 - 2x + 2x^2, p3(x) = 2 - 2x + 2x^2 - 2x^3, and p4(x) = 2 - 2x + 2x^2 - 2x^3 + 2x^4`

. The nth Maclaurin polynomial in sigma notation is pn

`(x) = sum (2(-x)^m)/m!`

from m=0 to n.

Step-by-step explanation:

The student is asking how to find the Maclaurin polynomials of orders n=0, 1, 2, 3, and 4 for the function 2/(1+x), and also for the general nth Maclaurin polynomial in sigma notation. To find these, we use the fact that the Maclaurin series expansion of a function f(x) is given by

`f(x) = f(0) + x(f'(0))/1! + (x^2)(f''(0))/2! + ... + (x^n)(f^n(0))/n!`

where

`f^n(0)`

is the nth derivative of f evaluated at x=0.

For the function 2/(1+x), the Maclaurin series is a geometric series with a common ratio of -x, which converges for |x|<1. The first few terms are calculated using the derivatives of 2/(1+x), resulting in polynomial approximations of increasing order:

• 
  `p0(x) = 2p1(x) = 2 - 2xp2(x) = 2 - 2x + 2x^2p3(x) = 2 - 2x + 2x^2 - 2x^3p4(x) = 2 - 2x + 2x^2 - 2x^3 + 2x^4`

The general nth Maclaurin polynomial in sigma notation can be written as:

`pn(x) = ∑ (2(-x)^m)/m!`

from m=0 to n