This week, we’ll work out some Taylor Series expansions of popular functions. \(f(x)= \frac{1}{1−x}\\f(x) = e^x\\f(x) = ln(1+x)\\f(x)=x^{\frac{1}{2}}\)
For each function, only consider its valid ranges as indicated in the notes when you are computing the Taylor Series expansion. Please submit your assignment as an R- Markdown document.
library(calculus)
function_1=function(x) { (1-x)^-1}
taylor(function_1, var=c(x=0), order=6)## $f
## [1] "(1) * 1 + (0.999999999999872) * x^1 + (0.999999999868096) * x^2 + (0.999994567806407) * x^3 + (0.999693345452476) * x^4 + (0.976915998461401) * x^5 + (0.840820581190503) * x^6"
##
## $order
## [1] 6
##
## $terms
## var coef degree
## 0 1 1.0000000 0
## 1 x^1 1.0000000 1
## 2 x^2 1.0000000 2
## 3 x^3 0.9999946 3
## 4 x^4 0.9996933 4
## 5 x^5 0.9769160 5
## 6 x^6 0.8408206 6\(f(x)=\frac{1}{1-x}\\f'(x)=\frac{1}{(1-x)^2}\\f''(x)=\frac{2}{(1-x)^3}\\f'''(x)=\frac{6}{(1-x)^4}\\f''''(x)=\frac{24}{(1-x)^5}\\ \text{Let}\space a=0\\f(0)=1\\f'(0)=1\\f''(0)=2\\f'''(0)=6\\f''''(0)=24\\1+\frac{1}{1!}x+\frac{2}{2!}x^2+\frac{6}{3!}x^3+\frac{24}{4!}x^4\\1+x+x^2+x^3+x^4...\\=\sum\limits_{n=0}^{\infty}x^n\\\)
function_2=function(x) {exp(x)}
taylor(function_2, var=c(x=0), order=6)## $f
## [1] "(1) * 1 + (1.00000000000369) * x^1 + (0.49999999999841) * x^2 + (0.166666665586208) * x^3 + (0.0416666593058216) * x^4 + (0.00833328287074409) * x^5 + (0.00138886379618681) * x^6"
##
## $order
## [1] 6
##
## $terms
## var coef degree
## 0 1 1.000000000 0
## 1 x^1 1.000000000 1
## 2 x^2 0.500000000 2
## 3 x^3 0.166666666 3
## 4 x^4 0.041666659 4
## 5 x^5 0.008333283 5
## 6 x^6 0.001388864 6\(f(x)=e^x\\f'(x)=f''(x)=f'''(x)=f''''(x)=e^x\\\text{Let}\space a=0\\f(0)=f'(0)=f''(0)=f'''(0)=f''''(0)=1\\1+\frac{1}{1!}x+\frac{1}{2!}x^2+\frac{1}{3!}x^3+\frac{1}{4!}x^4\\1+x+\frac{1}{2}x^2+\frac{1}{6}x^3+\frac{1}{24}x^4...\\=\sum\limits_{n=0}^{\infty}\frac{x^n}{n!}\\\)
function_3=function(x) {log(x+1)}
taylor(function_3, var=c(x=0), order=6)## $f
## [1] "(0.999999999994524) * x^1 + (-0.499999999988448) * x^2 + (0.333332558064701) * x^3 + (-0.249961920877318) * x^4 + (0.19754431053083) * x^5 + (-0.152263325753268) * x^6"
##
## $order
## [1] 6
##
## $terms
## var coef degree
## 0 1 0.0000000 0
## 1 x^1 1.0000000 1
## 2 x^2 -0.5000000 2
## 3 x^3 0.3333326 3
## 4 x^4 -0.2499619 4
## 5 x^5 0.1975443 5
## 6 x^6 -0.1522633 6\(f(x)=ln(1+x)\\f'(x)=\frac{1}{1+x}\\f''(x)= \frac{-1}{(1+x)^2}\\f'''(x)=\frac{2}{(1+x)^3}\\f''''(x)=\frac{-6}{(1+x)^4}\\\text{Let}\space a=0\\f(0)=0\\f'(0)=1\\f''(0)=-1\\f'''(0)=2\\f''''(0)=-6\\0+\frac{1}{1!}x+\frac{-1}{2!}x^2+\frac{2}{3!}x^3+\frac{-6}{4!}x^4\\0+x-\frac{1}{2}x^2+\frac{1}{3}x^3-\frac{1}{4}x^4...\\=\sum\limits_{n=1}^{\infty}(-1)^{n-1}\frac{x^n}{n}\\\)
function_4=function(x) x^0.5
taylor(function_4, var=c(x=1), order=6)## $f
## [1] "(1) * 1 + (0.499999999994206) * (x-1)^1 + (-0.125000000001415) * (x-1)^2 + (0.0624999125848509) * (x-1)^3 + (-0.0390585237861548) * (x-1)^4 + (0.0271073978028002) * (x-1)^5 + (-0.019231950220526) * (x-1)^6"
##
## $order
## [1] 6
##
## $terms
## var coef degree
## 0 1 1.00000000 0
## 1 (x-1)^1 0.50000000 1
## 2 (x-1)^2 -0.12500000 2
## 3 (x-1)^3 0.06249991 3
## 4 (x-1)^4 -0.03905852 4
## 5 (x-1)^5 0.02710740 5
## 6 (x-1)^6 -0.01923195 6\(f(x)=x^{\frac{1}{2}}\\f'(x)=\frac{1}{2}x^{-\frac{1}{2}}\\f''(x)= -\frac{1}{4}x^{-\frac{3}{2}}\\f'''(x)=\frac{3}{8}x^{-\frac{5}{2}}\\f''''(x)=-\frac{15}{16}x^{-\frac{7}{2}}\\\text{Let}\space a=1\\f(1)=1\\f'(1)=\frac{1}{2}\\f''(1)=-\frac{1}{4}\\f'''(1)=\frac{3}{8}\\f''''(1)=-\frac{15}{16}\\1+\frac{\frac{1}{2}}{1!}(x-1)+\frac{-\frac{1}{4}}{2!}(x-1)^2+\frac{\frac{3}{8}}{3!}(x-1)^3+\frac{-\frac{15}{16}}{4!}(x-1)^4\\1+\frac{1}{2}(x-1)-\frac{1}{8}(x-1)^2+\frac{1}{16}(x-1)^3-\frac{5}{128}(x-1)^4...\\\)