proof of alternating series test
The series has partial sum
where the ’s are all nonnegative and nonincreasing. From above, we have the following:
Since , we have . Moreover,
Because the ’s are nonincreasing, we have for any . Also, . Thus, . Hence, the even partial sums and the odd partial sums are bounded. Also, the even partial sums ’s are monotonically nondecreasing, while the odd partial sums ’s are monotonically nonincreasing. Thus, the even and odd series both converge.
We note that . Therefore, the sums converge to the same limit if and only if as . The theorem is then established.
|Title||proof of alternating series test|
|Date of creation||2014-07-22 16:20:39|
|Last modified on||2014-07-22 16:20:39|
|Last modified by||pahio (2872)|