limit theorems and series
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By using partial fractions show that
a. the sumation from 0 to infinity of 1/(n+1)(n+2)=1
b. the sumation from 0 to infinity of 1/(alpha+n)(alpha+n+1)=1/alpha >0 if alpha>0
c. the sumation from 0 to infinity of 1/n(n+1)(n+2) =1/4
apply the theorem: let (Xsubn) be a sequence of positive real numbers such that L := lim(Xsubn+1/ Xsubn) exsists. If L>0 then (Xsubn) converges and lim(Xsubn)=0
apply this theorem where a,b satisfy 0 < a < 1, b > 1
a. (a^n)
b.(b^n/2^n)
c.(n/b^n)
d.(2^3n/3^2n)
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Solution Summary
This solution is comprised of a detailed explanation to show that
a. the sumation from 0 to infinity of 1/(n+1)(n+2)=1
b. the sumation from 0 to infinity of 1/(alpha+n)(alpha+n+1)=1/alpha >0 if alpha>0
c. the sumation from 0 to infinity of 1/n(n+1)(n+2) =1/4
Solution Preview
Problem #1
a. Since , then we have
b. Since , then we have
c. Since , then we have
Problem #2
The theorem is: ...
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