Enter the nth term of the first series and the nth term of the second series into the calculator to determine the convergence or divergence of the series. This calculator can also evaluate any of the variables given the others are known.
Direct Comparison Test Formula
The following formula is used to calculate the Direct Comparison Test.
If 0 ≤ a_n ≤ b_n for all n and ∑b_n converges, then ∑a_n also converges. If 0 ≤ a_n ≤ b_n for all n and ∑a_n diverges, then ∑b_n also diverges.Variables:
- a_n is the nth term of the first series b_n is the nth term of the second series ∑a_n is the sum of the first series ∑b_n is the sum of the second series
To calculate the Direct Comparison Test, first ensure that the nth term of the first series is less than or equal to the nth term of the second series for all n. If the sum of the second series converges and the inequality holds, then the sum of the first series also converges. If the sum of the first series diverges and the inequality holds, then the sum of the second series also diverges.
What is a Direct Comparison Test?
The Direct Comparison Test is a method used in calculus to determine whether a series converges or diverges. It involves comparing the series in question to another series that is already known to converge or diverge. If the series is less than a converging series, then it also converges. Conversely, if the series is greater than a diverging series, then it also diverges. This test is particularly useful when dealing with series that are difficult to evaluate directly.
How to Calculate Direct Comparison Test?
The following steps outline how to calculate the Direct Comparison Test.
- First, determine the nth term of the first series, a_n.
- Next, determine the nth term of the second series, b_n.
- Next, calculate the sum of the first series, ∑a_n.
- Then, calculate the sum of the second series, ∑b_n.
- Finally, compare the values of ∑a_n and ∑b_n to determine if the series converge or diverge.
Example Problem :
Use the following variables as an example problem to test your knowledge.
nth term of the first series, a_n = 1/n^2
nth term of the second series, b_n = 1/n
sum of the first series, ∑a_n = ∑(1/n^2)
sum of the second series, ∑b_n = ∑(1/n)
