Direct Comparison Test Calculator

To use the Direct Comparison Test, compare the terms of two series. If $0 \leq a_n \leq b_n$ for all $n$ and $\sum b_n$converges, then $\sum a_n$ also converges. Similarly, if $0 \leq a_n \leq b_n$for all $n$ and $\sum a_n$ diverges, then $\sum b_n$ also diverges.

 

Direct Comparison Test Calculator

Enter any 2 values to calculate the missing variables

nth term of the first series (a_n) nth term of the second series (b_n) Sum of the first series (∑a_n) Sum of the second series (∑b_n)

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The Direct Comparison Test Calculator assists you to calculate whether a series converges or diverges by comparing it with another known series. In essence, this test is a powerful tool used in calculus to evaluate series whose behavior may be difficult to assess directly.

For instance, if you know that a series with larger terms converges, and your series has smaller terms, you can conclude that your series also converges. This comparison is vital for understanding the convergence or divergence of series in mathematical analysis.

Formula

If $0 \leq a_n \leq b_n$  for all $n$ and $\sum b_n$ converges, then $\sum a_n$ also converges.
If $0 \leq a_n \leq b_n$ for all $n$ and $\sum a_n$ diverges, then $\sum b_n$ also diverges.

Solved Calculations

Example 1:
Given that $0 \leq a_n \leq b_n$ and $\sum b_n$ converges, we can conclude that $\sum a_n$ also converges.

Step	Calculation
$0 \leq a_n \leq b_n$	True for all $n$
$\sum b_n$ converges	Known fact
Conclusion	$\sum a_n$ converges

Answer: The series $\sum a_n$ converges.

Example 2:
If $0 \leq a_n \leq b_n$ for all $n$ and $\sum a_n$ diverges, then we can conclude that $\sum b_n$ also diverges.

Step	Calculation
$0 \leq a_n \leq b_n$	True for all $n$
$\sum a_n$ diverges	Known fact
Conclusion	$\sum b_n$diverges

Answer: The series $\sum b_n$ diverges.

What is the Direct Comparison Test Calculator?

The Direct Comparison Test Calculator is an excellent tool. It is used in calculus to calculate the convergence or divergence of infinite series. The direct comparison test helps compare a given series with a known, simpler series to establish its behavior.

By using this test, you can determine if a series converges or diverges based on whether it is smaller or larger than a convergent series.

To use the calculator, input the series you want to test, and it will compare it to a known series. If the given series behaves similarly to the known series, the calculator will help you conclude whether the series converges or diverges. This method is especially useful for series that are difficult to analyze directly.

Final Words:

In conclusion, the Direct Comparison Test Calculator is an efficient and effective tool for solving series problems in calculus. It simplifies the comparison process, helping you determine the convergence or divergence of series with ease.