Power Series and Exact Values of Numerical Series

Key Questions

• How do you use a power series to find the exact value of the sum of the series #pi-pi^2/2+pi^4/(4!) -pi^6/(6!) + …# ?

  Since

  #1-x^2/{2!}+{x^4}/{4!}-{x^6}/{6!}+cdots=cosx#,

  #1-pi^2/{2!}+{pi^4}/{4!}-{pi^6}/{6!}+cdots=cos(pi)=-1#

  Therefore,

  #pi-pi^2/{2!}+{pi^4}/{4!}-{pi^6}/{6!}+cdots#

  #=pi-1+(1-pi^2/{2!}+{pi^4}/{4!}-{pi^6}/{6!}+cdots)#

  #=pi-1+(-1)=pi-2#

  I hope that this was helpful.

  Wataru · · Sep 28 2014
• How do you use a power series to find the exact value of the sum of the series #1-1/2+1/3-1/4 + …# ?

  Alternating Harmonic Series

  #sum_{n=1}^infty(-1)^{n-1}/n=1-1/2+1/3-1/4+cdots=ln2#

  Since

  #ln(1-x)=-sum_{n=1}^infty{x^n}/n#,

  by setting #x=-1#,

  #ln2=-sum_{n=1}^infty{(-1)^n}/n=sum_{n=1}^infty(-1)^{n-1}/n#

  Wataru · · Sep 25 2014
• How do you use a power series to find the exact value of the sum of the series #1+2+4/(2!) +8/(3!) +16/(4!) + …# ?

  Since

  #1+x+x^2/{2!}+x^3/{3!}+x^4/{4!}+cdots=e^x#,

  by replacing #x# by #2#,

  #1+2+2^2/{2!}+2^3/{3!}+2^4/{4!}+cdots=e^2#.

  I hope that this was helpful.

  Wataru · 1 · Oct 10 2014

• How do you use a power series to find the exact value of the sum of the series #1+2+4/(2!) +8/(3!) +16/(4!) + …# ?
• How do you use a power series to find the exact value of the sum of the series #1-1/2+1/3-1/4 + …# ?
• How do you use a power series to find the exact value of the sum of the series #pi-pi^2/2+pi^4/(4!) -pi^6/(6!) + …# ?
• How do you use a power series to find the exact value of the sum of the series #1-pi^2/(2!)+pi^4/(4!) +pi^6/(6!) + …# ?
• How do you use a power series to find the exact value of the sum of the series #1+sqrt(2)+2/(2!) +(sqrt(2))^3/(3!) +4/(4!) + …# ?
• How do you use a power series to find the exact value of the sum of the series #1+e+e^2 +e^3 +e^4 + …# ?
• How do you use a power series to find the exact value of the sum of the series #pi/4-(pi/4)^3/(3!)+(pi/4)^5/(5!)-(pi/4)^7/(7!) + …# ?
• How do you use a power series to find the exact value of the sum of the series #1-(pi/4)^2/(2!)+(pi/4)^4/(4!) -(pi/4)^6/(6!) + …# ?