Introduction to Power Series

Key Questions

• How do you find the power series for a function centered at #c# ?

  Taylor Series centered at c#

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

  I hope that this was helpful.

  Wataru · · Oct 10 2014
• What are the most important power series to memorise?

  Useful Maclaurin Series

  #1/{1-x}=sum_{n=0}^infty x^n#

  #e^x=sum_{n=0}^infty{x^n}/{n!}#

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

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

  ---

  I hope that this was helpful.

  Wataru · · Oct 19 2014
• What is a Power Series?

  You can thing of a power series as a polynomial function of infinite degree since it looks like this:

  #sum_{n=0}^inftya_nx^n=a_0+a_1x+a_2x^2+a_3x^3+cdots#

  I hope that this was sufficient.

  Wataru · · Sep 26 2014

• What is a Power Series?
• How do you find the power series for a function centered at #c# ?
• How do you test a power series for convergence?
• How do you find the radius of convergence for a power series?
• How do you find the interval of convergence for a power series?
• How do you find a power series representation for #f(x)=1/(1+x)^3#?
• How do you find a power series representation for # x^2 / ( 1 - 2x )^2#?
• How do you find a power series representation for # (1+x)/((1-x)^2)#?
• How do you find a power series representation for #(x-2)^n/(n^2) # and what is the radius of convergence?
• How do you find a power series representation for #f(x)=3/((1-5x)^2)# and what is the radius of convergence?
• How do you find a power series representation for #x/(1-x^2)# and what is the radius of convergence?
• How do you find a power series representation for #e^(-x^2)# and what is the radius of convergence?
• How do you find a power series representation for #f(x)= 1/(1+4x^2)# and what is the radius of convergence?
• How do you find a power series representation for #f(x) = x / (1+x^2) # and what is the radius of convergence?
• How do you find a power series representation for #f(x)= 1/(1+x)# and what is the radius of convergence?
• How do you find a power series representation for #f(x)= x/(9+x^2)# and what is the radius of convergence?
• How do you find a power series representation for #x^3/(2-x^3)# and what is the radius of convergence?
• How do you find a power series representation for #(1+x)/(1-x) # and what is the radius of convergence?
• How do you find a power series representation for #ln(1-x^2) # and what is the radius of convergence?
• How do you find a power series representation for #x^2 arctan(x^3# and what is the radius of convergence?
• How do you find a power series representation for #f(x)= 1/(3-x)# and what is the radius of convergence?
• How do you find a power series representation for #e^x# and what is the radius of convergence?
• How do you find a power series representation for #f(x) = ln(x^2+1)# and what is the radius of convergence?
• How do you find a power series representation for #ln(8-x)# and what is the radius of convergence?
• How do you find a power series representation for #f(x) = (x) / ((1-x)^2)# and what is the radius of convergence?
• How do you find a power series representation for #(arctan(x))/(x) # and what is the radius of convergence?
• How do you find a power series representation for #ln(1-x)# and what is the radius of convergence?
• How do you find a power series representation for # f(x)=(x / (x^(2)-3x+2) )# and what is the radius of convergence?
• How do you find a power series representation for # (10x)/(14+x) # and what is the radius of convergence?
• How do you find a power series representation for #f(x)=xln(x+1)# and what is the radius of convergence?
• How do you find a power series representation for # f(x)= 9/(1-3x)^2 # and what is the radius of convergence?
• How do you find a power series representation for #ln(5-x) # and what is the radius of convergence?
• How do you find a power series representation for # (x/(2-x))^3# and what is the radius of convergence?
• How do you find a power series representation for # f(z)=z^2 # and what is the radius of convergence?
• How do you find a power series representation for #1/(1-x)^2 # and what is the radius of convergence?
• How do you use a power series to approximate # int_0^1xtan^-1xdx #?
• Question #d27e0
• What is the power series representation of #ln((1+x)/(1-x))#?
• What is the power series for #ln(1+x^4)#?
• How do you find the power series of ln(1+x)?
• The exponential function #e^x# can be defined as a power series as: #e^x=sum_(n=0)^oo x^n/(n!)=1+x+x^2/(2!)+x^3/(3!)+...# Can you use this definition to evaluate #sum_(n=0)^(oo)((0.2)^n e^-0.2)/(n!)#?
• How do you calculate Euler's Number?
• What is the Maclaurin Series for #tanax#?