Let an be a decreasing sequence with limit o Let L and sn de

Let {a_n} be a decreasing sequence with limit o. Let L and s_n denote, respectively, the sum and nth partial sum of the series sigma^infinity_n = 1 (-1)^n +1 a_n. Prove that S_n - L

Solution

To start with , the sum L makes sense because of Leibniz\'s theorem:

Note that all a_n are positive as it is a decreasing sequence., tending to 0

Now |S_n -L|= |a_n+1 -a_n+2+a_n+3-a_n+4....|

                  = a_n+1 -(a_n+2-a_n+3)-(a_n+4-a_n+5)...

                 < a_n+1 , as each term in the bracket is positive , because the sequence decreases