How to Apply Comparison, Ratio, and Root Tests

Step-by-step Guide to Applying Comparison, Ratio, and Root Tests

Here is a step-by-step guide to applying comparison, ratio, and root tests:

Step 1: Introduction Comparison Test

Purpose: This test often comes into play when we’re given a series whose terms remind us of another series that we already understand.

Step-by-Step:

1. Identify the Target Series: You have a series, \(∑a_n\)​, whose convergence or divergence you wish to ascertain.
2. Select a Comparator Series: Identify another series, \(∑b_n\)​, whose convergence/divergence is known.
3. Inequality Condition: Check if:
   • \(0≤a_n​≤b_n\)​ for all \(n\), or
   • \(0≤b_n​≤a_n\)​ for all \(n\).
4. Deduction:
   • If \(∑b_n\)​ converges and you have \(0≤a_n​≤b_n\)​​, then \(∑a_n\)​ also converges.
   • If \(∑b_n\)​​ diverges and you have \(0≤b_n​≤a_n\)​, then \(∑a_n\)​ also diverges.

Step 2: Ratio Test

Purpose: Applicable when considering series involving factorials, exponentials, or powers. This test compares the ratio of consecutive terms.

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Step-by-Step:

1. Calculate the Ratio: Compute the limit: \(L=lim_{n→∞​}​|\frac{a_{n+1}}{a_n}|\)
2. Convergence Criteria:
   • If \(L<1\), then the series \(∑a_n​\) is absolutely convergent, hence convergent.
   • If \(L>1\), then the series \(∑a_n\)​ is divergent.
   • If \(L=1\), the test is inconclusive.

Step 3: Root Test

Purpose: This test examines the nth roots of the terms of a series and is particularly useful for series where terms are raised to the power of n.

Step-by-Step:

1. Calculate the nth Root: Compute the limit: \(L=lim_{n→∞​}​|{a_n}|^{\frac{1}{n}}\)
2. Convergence Criteria:
   • If \(L<1\), then the series \(∑a_n\)​ converges absolutely, hence converges.
   • If \(L>1\), then the series \(∑a_n​\) diverges.
   • If \(L=1\), the test is inconclusive.

Final Word:

Working through the vast sea of infinite series, the Introduction Comparison Test, Ratio Test, and Root Test serve as lighthouses, illuminating paths to understanding convergence or divergence. While each test has its unique strengths and applications, mastery over them provides a solid toolkit for any mathematical voyage into the world of series. Embrace the journey and may your series always converge to enlightenment!