determine if the sequence is increasing, decreasing, or not monotonic. a((1x3x5x...x(2^n-1))/(2^n x n!)?

Question

x means times

Answer 1

Look at the ratio of the (n+1)-st term to the n-th. After cancellation, you have [(2^n + 1)*(2^n + 3)*...*(2^(n+1) - 1)]/[2*(n + 1)] which is > 1 when n >= 2. Therefore, a_(n+1) > a_n, so the sequence is eventually increasing.