Wednesday, 20 April 2016

summation - sum of this series: suminftyn=1frac14n21




I am trying to calculate the sum of this infinite series after having read the series chapter of my textbook: n=114n21



my steps:



n=114n21=n=124n21n=114n21=..help..=sum



I am lacking some important properties, I feel I am coming to the right step and cannot spit that out..


Answer



Hint: Partial Fraction decomposition:14n21=1(2n1)(2n+1)=12[12n112n+1]



You must then compute the closed form of
kn=1[12n112n+1]

Can you do that?
Note that
kn=112n1=11+13+...+12k1=120+1+121+1+...+12(k1)+1=k1n=012n+1=kn=112n+1+120+112k+1


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