have an account?【sign in】proof of alternating series test: Proving Alternating Series Tests through Mathematical Induction
bangsauthorThe alternating series test is a useful tool in analyzing series with alternating terms in positive and negative signs. This test allows us to easily determine whether a series is convergent or divergent. In this article, we will explore the alternating series test and prove it through mathematical induction.
Alternating Series and the Alternating Series Test
An alternating series is a series where terms alternate between positive and negative signs. For example, consider the series:
a + (-b) + (-c) + (-d) + (e) + (+f) + (+g) + (-h) + (+i) + (-j) + ...
This series contains terms with alternating signs, and can be easily analyzed using the alternating series test. The alternating series test states that a series is convergent if the sum of the first n terms is less than some constant multiple of n, and divergent if the sum of the first n terms is greater than some constant multiple of n for some n.
Proof through Mathematical Induction
To prove the alternating series test, we will use mathematical induction. We begin with the base case, n = 1, and then prove the result for higher values of n.
Base Case: n = 1
For the base case, we have one term in the series:
a + (-b) + (-c) + (-d) + (e) + (+f) + (+g) + (-h) + (+i) + (-j) + ...
We know that the sum of the first one term is:
a - b = a
Since a - b k * n
Since the sum of the first n terms is less than k * n, and the sum of the first (n + 1) terms is greater than k * n, the series is divergent for n + 1.
Therefore, we have proven that the alternating series test is true for n + 1.
Through mathematical induction, we have proven the alternating series test. This test allows us to easily determine whether a series with alternating terms is convergent or divergent. By starting with the base case and proving the result for higher values of n, we can easily extend the result to series with any number of terms. This tool is particularly useful in analyzing complex series with multiple terms and can help us better understand the behavior of these series.