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?
To determine if the series converges or diverges using the alternating series test, there are two conditions that need to be satisfied:
1. The series must be alternating. That means the terms of the series must alternate in sign, switching between positive and negative.
2. The absolute value of the terms must decrease as n increases, meaning that the terms must become smaller in magnitude.
Let's examine the series you provided:
Sum from n=1 to infinity of (-1)^n/(3n+1)
1. First, we need to check if the series is alternating. In this case, the series alternates between positive and negative because of the term (-1)^n. So, the first condition is satisfied.
2. Next, we need to check if the absolute value of the terms is decreasing. To do this, we can look at the ratio of the consecutive terms:
|((-1)^(n+1))/(3(n+1)+1) / ((-1)^n)/(3n+1)|
Simplifying this expression gives:
|(-1)^(n+1)(3n+1) / (-1)^n(3(n+1)+1)|
The (-1)^(n+1) term cancels out, resulting in:
|(3n+1) / (3(n+1)+1)|
To determine if this ratio is less than 1 for all n, we can simplify further:
|(3n+1) / (3n+4)|
As n increases indefinitely, the ratio approaches 1/3. Since 1/3 is less than 1, the second condition is satisfied.
Since both conditions of the alternating series test are met, the series converges. Therefore, you are correct in saying that the series converges.