Proving Raabe's Criteria for Convergent Sums: Steps and Hints

[MathematicalPhysicist]

i'm having a problem proving raabe's criteria for convergent sums.
here in planetmath there's a description of it:
http://planetmath.org/encyclopedia/RaabesCriteria.html

i got that the first inequality is correct when mu is smaller than 1.

i got a hint in my test that i should show that {na_(n+1)} is monotonely decreasing, which i did and by another theorem to show c_n=(n-1)a_n-na_(n+1) is convergent which i also did, but i got that
(1-a_(n+1)/a_n)*n=(1-(1-1/n))*n=1>=mu.
where have i gone wrong?

thanks in advance.

 

[greytomato]

a Mathematical series is shown in this fasion:
a_(n+1) = a_n * q
if |q|<1 then the series is convergent.

in your case, you see than q = a_(n+1)/a_n
all you really need to show is that q<1.

remember:
the sum of that series = (1 - q^n)/(1-q)
when n goes to infinity then the sum = 1/(1-q)

 

[Muzza]

greytomato, no. Your "q" must be constant. Using your logic, the harmonic series would be convergent...

 

[MathematicalPhysicist]

so muzza, what approach should i take here?