Would it be possible for someone to fully breakdown this Ser

Would it be possible for someone to fully breakdown this Series question step by step including the steps in applying the test? Thank you

Solution

a) This is convergent because the ratio is 3/4 and |3/4| < 1

To show the ratio is 3/4

^{r = (-1) ^(n+1) (-3)^(n+1)/2^(2(n+1)) / ((-1)^n (-3)^n/2^(2n) ) =}

^{(-1)^n -1 (-3)^n (-3)/2^2n/2^2 / ((-1)^n (-3)^n/2^2(n)) =}

^{-1(-3)/2^2 =}

^3/4

^{b) The sequence does not have a limit of 0, so the sum does not converge.}

^{This is because the largest degree in both the numerator and denominator is 3, and the coefficient on the cubic term in the numerator is 1 and the coefficient on the cubic term in the denominator is 8. The (-1)^n term is what makes us have a lim sup of 1/8 and a lim inf of -1/8.}