Use the alternating series test to determine the convergence/divergence of the series the summation from n equals 1 to infinity of the product of negative 1 raised to the nth power and the...

Question

Use the alternating series test to determine the convergence/divergence of the series  

A	The alternating series fails.
B	The series converges.
C	The series is conditionally convergent.
D	The summation must start with n = 0 so the alternating series test cannot be applied.

Answer 1

The answer is A. For the alternating series test to prove convergence, two elements must be present:

1) The lim n goes to infinity of Bn = 0 (note Bn is the absolute value of An shown here)

2) Ultimately, Bn must be decreasing

This series fails both. The lim n goes to oo Bn = 3/4 (not 0) and

if you take derivative of function f(x) = 3x/(4x-1) you get 3/(4x-1) + 12x/(4x-1)^2 which is positive for large values of X meaning that the function is increasing where the requirement is to decrease.