test the series for convergence or divergence using the alternating series test
the sum from n=1 to infinity of (-1)^n/(3n+1)
I said it converges, is this true?
Hi:
You are correct. The test criteria are,
1) The terms a_n are of decreasing sequence; clearly 1/(3n+1) is decreasing on [1, inf].
2) Limit(n->inf)[a_n] = 0.
Both criteria having been met, we conclude convergence.
Regards,
Rich B.
"If the sequence An converges to 0, and each An is smaller than An-1 (i.e. the sequence An is monotone decreasing), then the series converges."
In this case, An = 1 / (3n + 1)
The limit as n->infinity (An) = 0
A1 = 1 / 4
A2 = 1 / 7
Therefore, by the Alternating Series Test the series converges.
To test the convergence of the given series using the alternating series test, we need to check two conditions:
1. The absolute value of the terms must decrease as n increases.
2. The limit of the terms as n approaches infinity must be zero.
Let's examine these conditions for the given series:
1. The terms of the series are (-1)^n / (3n + 1). Note that the denominator, 3n + 1, is an increasing function of n. Since the absolute value of the terms does not decrease as n increases, this condition is not satisfied.
2. To check the limit of the terms, we can evaluate the limit as n approaches infinity. Taking the limit of (-1)^n / (3n + 1), the term oscillates between positive and negative values. As n approaches infinity, the absolute value of (-1)^n becomes 1. The denominator, 3n + 1, increases indefinitely, so the limit of the terms does not approach zero.
Based on these conditions, we can conclude that the given series diverges. Therefore, your initial statement that the series converges is incorrect.
To test the given series for convergence or divergence, you correctly applied the alternating series test. According to the alternating series test, an alternating series of the form ∑(-1)^(n-1)*an, where an is a positive number, converges if three conditions are fulfilled:
1. The absolute value of the terms {an} decreases as n increases, that is, |an+1| ≤ |an|.
2. The limit of the terms {an} approaches 0 as n approaches infinity, that is, lim(n→∞) an = 0.
3. The terms {an} are positive, meaning an > 0 for all n.
Let's apply these conditions to the given series:
The terms of the given series are {an} = 1/(3n+1).
1. To see if the absolute value of the terms decreases as n increases, we can take the derivative of the terms and check if it is negative. So, let's evaluate d(an)/dn = -3/(3n+1)^2. Since this derivative is negative for all positive values of n, we can conclude that the terms {an} decrease in absolute value as n increases. Condition 1 is satisfied.
2. To check if the limit of the terms approaches 0, we can evaluate the limit as n approaches infinity:
lim(n→∞) 1/(3n+1) = 0. Hence, condition 2 is satisfied.
3. The terms {an} = 1/(3n+1) are positive for all n. Hence, condition 3 is satisfied.
Since all three conditions of the alternating series test are met, we can conclude that the given series converges. Therefore, your answer that the series converges is indeed correct.