Constructing a Taylor Series

Key Questions

• How do you use a Taylor series to find the derivative of a function?

  One of the benefits of a Taylor series is the ease of differentiation since it can be done term by term. So, the Talyor series

  #f(x)=sum_{n=0}^infty{f^{(n)}(a)}/{n!}(x-a)^n#

  can be differentiated as

  #f'(x)=sum_{n=0}^infty{f^{(n)}(a)}/{n!}[(x-a)^{n}]'#

  by Power Rule,

  #=sum_{n=1}^infty{f^{(n)}(a)}/{n!}[n(x-a)^{n-1}]#

  (Note: #n# starts from #1# since the term is zero when #n=0#)
  by simplifying a little further,

  #=sum_{n=1}^infty{f^{(n)}(a)}/{(n-1)!}(x-a)^{n-1}#

  I hope that this was helpful.

  Wataru · · Sep 27 2014
• How do you use a Taylor series to solve differential equations?

  Let us solve #y''+y=0# by Power Series Method.

  Let #y=sum_{n=0}^inftyc_nx^n#, where #c_n# is to be determined.

  By taking derivatives,

  #y'=sum_{n=1}^inftync_nx^{n-1} Rightarrow y''=sum_{n=2}^inftyn(n-1)c_nx^{n-2}#

  We can rewrite #y''+y=0# as

  #sum_{n=2}^inftyn(n-1)c_nx^{n-2}+sum_{n=0}^inftyc_nx^n=0#

  by shifting the indices of the first summation by 2,

  #Rightarrow sum_{n=0}^infty(n+2)(n+1)c_{n+2}x^n+sum_{n=0}^inftyc_nx^n=0#

  by combining the summations,

  #Rightarrow sum_{n=0}^infty[(n+2)(n+1)c_{n+2}+c_n]x^n=0#,

  #Rightarrow (n+2)(n+1)c_{n+2}+c_n=0#

  #Rightarrow c_{n+2}=-{c_n}/{(n+2)(n+1)}#

  Let us look at even coefficients.

  #c_2={-c_0}/{2cdot1}=-c_0/{2!}#
  #c_4={-c_2}/{4cdot3}={-1}/{4cdot3}cdot{-c_0}/{2!}=c_0/{4!}#
  #c_6={-c_4}/{6cdot5}={-1}/{6cdot5}cdot c_0/{4!}=-{c_0}/{6!}#
  .
  .
  .
  #c_{2n}=(-1)^n{c_0}/{(2n)!}#

  Let us look at odd coefficients.

  #c_3={-c_1}/{3cdot2}=-{c_1}/{3!}#
  #c_5={-c_3}/{5cdot4}={-1}/{5cdot4}cdot{-c_1}/{3!}={c_1}/{5!}#
  #c_7={-c_5}/{7cdot6}={-1}/{7cdot6}cdot{c_1}/{5!}=-{c_1}/{7!}#
  .
  .
  .
  #c_{2n+1}=(-1)^n{c_1}/{(2n+1)!}#

  Hence, the solution can be written as:

  #y=sum_{n=0}^inftyc_nx^n#

  by splitting into even terms and odd terms,

  #=sum_{n=0}^inftyc_{2n}x^{2n}+sum_{n=0}^inftyc_{2n+1}x^{2n+1}#

  by pluggin in the formulas for #c_{2n}# and #c_{2n+1}# we found above,

  #=c_0sum_{n=0}^infty(-1)^n{x^{2n}}/{(2n)!}+c_1 sum_{n=0}^infty(-1)^n{x^{2n+1}}/{(2n+1)!}#

  by recognizing the power series,

  #=c_0 cosx+c_1 sinx#

  ---

  I hope that this was helpful.

  Wataru · · Oct 17 2014
• What is a Taylor series?

  The Taylor series of a function is a power series, all of whose derivatives match their corresponding derivatives of the function.

  Let us derive the Taylor series of a function #f(x)#, centered at #c#.

  Let

  #f(x)=sum_{n=0}^infty a_n(x-c)^n#

  #=a_0+a_1(x-c)+a_2(x-c)^2+cdots#,

  where coefficients #a_1, a_2, a_3,...# are to be determined.

  By taking the derivatives,

  #f'(x)=a_1+2a_2(x-c)+3a_3(x-c)^2+cdots#

  #f''(x)=2a_2+3cdot2 a_3(x-c)+4cdot3 a_4(x-c)^2+cdots#

  #f'''(x)=3cdot2 a_3+4cdot3cdot2a_4(x-c)+5cdot4cdot3a_5(x-c)^2+cdots#
  .
  .
  .

  By plugging in #x=c#,

  #f(c)=a_0=0! cdot a_0#

  #f'(c)=a_1=1! cdot a_1#

  #f''(c)=2a_2=2! cdot a_2#

  #f'''(c)=3cdot2 a_3=3! cdot a_3#
  .
  .
  .
  #f^{(n)}(c)=n! cdot a_n#

  By dividing by #n!#,

  #a_n={f^{(n)}(c)}/{n!}#

  Hence, we have the Taylor series of #f(x)#, centered at #c#

  #f(x)=sum_{n=0}^infty{f^{(n)}(c)}/{n!}(x-c)^n#.

  ---

  I hope that this was helpful.

  Wataru · · Oct 18 2014

• How do you find the Taylor series of #f(x)=1/x# ?
• How do you find the Taylor series of #f(x)=cos(x)# ?
• How do you find the Taylor series of #f(x)=e^x# ?
• How do you find the Taylor series of #f(x)=ln(x)# ?
• How do you find the Taylor series of #f(x)=sin(x)# ?
• How do you use a Taylor series to find the derivative of a function?
• How do you use a Taylor series to prove Euler's formula?
• How do you use a Taylor series to solve differential equations?
• What is the Taylor series of #f(x)=arctan(x)#?
• What is the linear approximation of #g(x)=sqrt(1+x)^(1/5)# at a =0?
• How do I approximate #sqrt(128)# using a Taylor polynomial centered at 125?
• What is a Taylor series approximation for #f(x)=cos(pix)# with n = 4 and a = 3?
• How do I construct a Taylor series for #f(x)=1/sqrt(x)# centered at x=4?
• What is an example of a Taylor Series?
• What are the first 3 nonzero terms in the Taylor series expansion about x = 0 for the function #f(x)=cos(4x)#?
• How do you approximate the #sqrt(128)# using a taylor polynomial centered at 0?
• How do you determine a taylor series approximation for #f(x)=cos(x)# where #n=4# and #a=3#?
• How do you find the quadratic taylor polynomial q(x,y) approximating #f(x,y)= e^(x) cos (5y)# about (0,0)?
• How do you find the first terms of the taylor series #f(x)=1/sqrt(x)# centered at #x=4#?
• How do you find the first five terms of the Taylor series for #f(x)=x^8+x^4+3# at #x=1#?
• How do you find the first three terms of the Taylor series for #f(x)=cos(5x)# for #x=0#?
• How do you write the taylor series for #f(x)=sqrt(x)# at #a=16# and find the radius of convergence.?
• How do you find the taylor series of #f(x)=sinx# at #a=pi/6#?
• How do you find the taylor series of #1 1x^2#?
• How do you find the coefficient of #x^6# in the taylor series expansion?
• How do you find the taylor series of #1 + sin(x)#?
• How to do taylor series expansion of #e^(-x^2"/2")#?
• What are some real life applications of using taylor series expansions?
• How do you do the Taylor series expansion for #f(x)=x/(1-x)# at a=1?
• What is the Taylor series of #e^((-x)^2)#?
• What is the Taylor series expansion of #f(x) = 1/x^2# at a=1?
• What is the Taylor Series Expansion for #sin(sin(x))#?
• How do you find the 1st 4 nonzero terms in the Taylor series expansion about x=0 for #f(x)= sqrt(1+x)#?
• How do you find the taylor series expansion of #f(x) =x/(1+x)# around x=0?
• What does the "a" means in the Taylor Series expansion?
• How do you use a Taylor series to expand: #f(x) = x^2 + 2x + 5# about x = 3?
• How do you do the taylor series expansion of #arctan(x)# and #xsinx#?
• How do you do the taylor expansion for #f(x)=log(x+1)# at x = 0?
• How do you do the taylor series for expanding #1/(2+x^2)#, about a = 0 up to order n = 2?
• What is the taylor series expansion for the tangent function (tanx)?
• How do you write the first four terms and the general term of the Taylor series expansion of #f(x) = 1/(x-1)# about x = 2?
• What is the taylor series of #xe^x#?
• What is the taylor expansion of #e^(-1/x)#?
• How do you find the first three terms of the Taylor's series for #sin(x^2) #?
• How do you find the first 3 non-zero terms in the Taylor series of the function #cosh(x)# about the point #x=ln8# and use this series to estimate cosh(2)?
• What is the Taylor series for #sin 2x#?
• How do you use Taylor series for #sin(x)# at #a = pi/3#?
• How do you find the taylor series for #y=(e^x)cos(x)#?
• How do you use Taylor series to estimate the accuracy of approximation for #f(x)=sqrt(x)# with #a=1# and #n=3# with #0.9<=x<=1.1#?
• What is the Taylor Series generated by #f(x) = x - x^3#, centered around a = -2?
• What is the taylor series for #x(e^(2x))#?
• What is the taylor series in sigma notation of: #sin^2(x)#?
• How do you find the third degree Taylor polynomial for #f(x)= ln x#, centered at a=2?
• How do you find the taylor series for #e^(x^2)#?
• How do you find the taylor series for #cos x#, #a=pi/3#?
• How do you write the Taylor series for #f(x)=coshx#?
• How do you find the taylor series of #sin (x^2)#?
• How do you find the Taylor series for #ln(x)# about the value x=1?
• How is the taylor series for #e^z# derived?
• What is the coefficient of x^2 in the Taylor series for #(1+x)^-2# about a=0?
• What is there in the derivation of the Taylor/Maclaurin series for #sin(x)# that determines if the series assumes #x# is radians or degrees?
• How do you find the taylor series for #f(x)=1/(1+x^3)#?
• How do you find the taylor series for #f(x)=xcos(x^2)#?
• How do you find the taylor series for #e^x - e^-x#?
• How do you find the taylor series for #f(x) = cos x # centered at a=pi?
• How do you find the taylor series series at x=0 of #f(x) = 1/(1-2x)#?
• How do you find the taylor series series at #a=-3# of #f(x) = x - x^3#?
• How do you find the taylor series series for # (4-x)^(1/2)#?
• How do you find the Taylor Series Expansion for #f(x)=e^(-3x)# at c = 2?
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• How do you find the taylor series for #(1/(1-x^2))# centered around #a = 3#?
• How do you find the taylor series series for #f(x)=lnx# at a=2?
• How do you find the taylor series series for #f (x) = 5 cos πx# at a=1/2?
• How do you find the taylor series series for #f(x) = x^4 - x^2 + 1# at c=-1?
• What is the taylor series used for?
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• How do you write the first 4 nonzero terms AND the general term of the taylor series for #e^((-x)^2)# centered at x=0?
• How do you write the first four nonzero terms and the general term of the taylor series for #f(x)= (2x)/ (1+x^2)# about x=0?
• How do you write the first four nonzero terms of the taylor series for #ln(1+x^2) #about x=0?
• How do you write the taylor series about 0 for the function #(1+x)^3#?
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• How can you find the taylor expansion of #sqrt (x) # about x=1?
• How can you find the taylor expansion of #ln(1-x)# about x=0?
• How can you find the taylor expansion of #1/(2+x^2)# about x=0?
• How can you find the taylor expansion of #f(x) = x^2 + 2x + 5# about x=3?
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• What is the Maclaurin series of #f(x) = sin^2(x)#?
• What is the Maclaurin series of #f(x) = cos(x)#?
• Let f be a continuous function: a) Find #f(4) if ∫_0^(x^2) f(t) dt = x sin πx# for all #x#. b) Find #f(4) if ∫_0^ f(x)t^2 dt = x sin πx# for all #x#?
• Find the first five terms of the Taylor series for #x^8+ x^4 +3 " at " x = 0#?
• How do you find the Taylor expansion of #e^(-1/x)#?
• How do you find the taylor series for ln(1+x)?
• How do you find the taylor series for #ln(1+x^2)#?
• How do you find the second degree taylor polynomial for f' about 4 and use it to approximate f'(4.3) given #f(x) = 7 - 3(x-4) + 5(x-4)^2 - 2(x-4)^3 + 6(x-4)^4#?
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• How do you find the Taylor polynomial Tn(x) for the function f at the number a #f(x)=sqrt(3+x^2)#, a=1, n=2?
• How do you find the Taylor polynomial of degree 4 for #f(x) = cosh(x)# about #x = 0 # by using the given Taylor polynomial for #e^x#?
• How do you find the Taylor polynomial for #1/(2-x)#?
• Given #f(x)= x^2 + 3x + 8# how do you determine the first three Taylor polynomials at x=0?
• How do you find the third degree Taylor Polynomial of #f(x)=ln(x^2)# at x=1?
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• How do you find the Taylor polynomials of orders 0, 1, 2, and 3 generated by f at a f(x) = cos(x), a= pi/4?
• How do you determine the third Taylor Polynomial at x=0 given #f(x)=cos(pi-5x)#?
• Suppose #T_4(x) = 7-3(x-2)+7(x-2)^2-6(x-2)^2+8(x-2)^4# is the degree 4 Taylor polynomial centered at x=2 for some function f, what is the value of f(2)?
• Suppose #T_4(x) = 7-3(x-2)+7(x-2)^2-6(x-2)^3+8(x-2)^4# is the 4th-degree Taylor polynomial centered at #x=2# for some function f, what is the value of #f^((3))(2)#?
• Suppose #T_4(x) = 7-3(x-2)+7(x-2)^2-6(x-2)^3+8(x-2)^4# is the degree 4 Taylor polynomial centered at x=2 for some function f, how do you estimate the value of f(2.1)?
• Suppose #T_4(x) = 7-3(x-2)+7(x-2)^2-6(x-2)^2+8(x-2)^4# is the degree 4 Taylor polynomial centered at x=2 for some function f, how do you estimate the value of f'(1.9)?
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