I am trying to calculate the sum of this infinite series after having read the series chapter of my textbook: ∞∑n=114n2−1
my steps:
∞∑n=114n2−1=∞∑n=124n2−1−∞∑n=114n2−1=..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:14n2−1=1(2n−1)(2n+1)=12[12n−1−12n+1]
You must then compute the closed form of
k∑n=1[12n−1−12n+1]
Can you do that?
Note that
k∑n=112n−1=11+13+...+12k−1=12⋅0+1+12⋅1+1+...+12(k−1)+1=k−1∑n=012n+1=k∑n=112n+1+12⋅0+1−12k+1
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