In Exercises 25 – 30, use the Taylor series given in Key Idea 8.8.1 to create the Taylor series of the given functions.
\(f(x)=(1+x)^\frac{1}{2}\cdot cos(x)\) (only find the first 4 terms)
| Function and Series | First Few Terms |
|---|---|
| \((1+x)^k=\sum_{n=1}^{\infty}\frac{k(k-1)\cdots (k-(n-1))}{n!}x^n\) | \(1+kx+\frac{k(k-1)}{2!}x^2+ \cdots\) |
| \(cos(x)=\sum_{n=1}^{\infty}(-1)^n \cdot \frac{x^{2n}}{(2n)!}\) | \(1- \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots\) |
\((1+x)^{\frac{1}{2}}=1+ \frac{1}{2}x+\frac{\frac{1}{2}(\frac{1}{2}-1)}{2!}x^2+ \frac{\frac{1}{2}(\frac{1}{2}-1)(\frac{1}{2}-2)}{2!}x^3+\cdots\) \(= 1+ \frac{x}{2} - \frac{x^2}{8}+ \frac{x^3}{16} + \cdots\) |
|---|
\(cos(x)=1- \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots\) \(=1- \frac{x^2}{2} + \frac{x^4}{24} - \frac{x^6}{720} + \cdots\) |
\(f(x)=(1+x)^\frac{1}{2}\cdot cos(x)\) \(=\left( 1+ \frac{1}{2}x - \frac{1}{8}x^2+ \frac{1}{16}x^3 + \cdots \right) \left (1+ \frac{x}{2} - \frac{x^2}{8}+ \frac{x^3}{16} + \cdots \right)\) \(= 1\cdot \left (1- \frac{x^2}{2} + \frac{x^4}{24} - \frac{x^6}{720} + \cdots \right ) + \frac{x}{2} \cdot \left (1- \frac{x^2}{2} + \frac{x^4}{24} - \frac{x^6}{720} + \cdots \right ) -\frac{x^2}{8} \cdot \left (1- \frac{x^2}{2} + \frac{x^4}{24} - \frac{x^6}{720} + \cdots \right ) +\frac{x^3}{16}\cdot \left (1- \frac{x^2}{2} + \frac{x^4}{24} - \frac{x^6}{720} + \cdots \right ) - \cdots\) \(=\left (1- \frac{x^2}{2} + \frac{x^4}{24} - \frac{x^6}{720} + \cdots \right)+ \left (\frac{x}{2} - \frac{x^3}{4} + \frac{x^5}{1440} - \frac{x^6}{1440}\cdots \right ) + \left (-\frac{x^2}{8} + \frac{x^4}{16} - \frac{x^6}{5760} + \frac{x^8}{5760}\cdots \right ) + \left( \frac{x^3}{16} -\frac{x^5}{32} + \frac{x^7}{11520}-\frac{x^9}{11520} \right )+\cdots\) \(=1 + \frac{x}{2} - \frac{5x^2}{8} - \frac{3x^3}{16} + \frac{23x^4}{360} - \frac{11x^5}{360} - \frac{x^6}{640} - \frac{7x^7}{11520} + \frac{x^8}{5760} - \frac{x^9}{11520} + \cdots\) |
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The R code below gives the first 10 terms of the taylor exapansion. The coefficients after fourth term do not match my calculations due to insufficient expansion.
library(calculus)
taylor("(x+1)^(1/2)*cos(x)", 0, var = "x", order = 9)## $f
## [1] "(1) * 1 + (0.5) * x^1 + (-0.625) * x^2 + (-0.1875) * x^3 + (0.0651041666666667) * x^4 + (0.0169270833333333) * x^5 + (-0.00757378472222222) * x^6 + (0.00435112847222222) * x^7 + (-0.00426732623387897) * x^8 + (0.0039183117094494) * x^9"
##
## $order
## [1] 9
##
## $terms
## var coef degree
## 0 1 1.000000000 0
## 1 x^1 0.500000000 1
## 2 x^2 -0.625000000 2
## 3 x^3 -0.187500000 3
## 4 x^4 0.065104167 4
## 5 x^5 0.016927083 5
## 6 x^6 -0.007573785 6
## 7 x^7 0.004351128 7
## 8 x^8 -0.004267326 8
## 9 x^9 0.003918312 9