How to Check If an Infinite Sum Converges?

Checking for the convergence of an infinite sum (series) involves several tests and considerations, as there isn't one single method that works for all cases. Here's a breakdown of common techniques:

1. Preliminary Check: The Divergence Test (nth Term Test)

• Description: This is the first and simplest test. If the limit of the terms of the series (a_n) as n approaches infinity is not zero, then the series diverges.

• Mathematically: If lim_n→∞ a_n ≠ 0, then ∑a_n diverges.

• Example: The series ∑ (n / (n + 1)) diverges because lim_n→∞ (n / (n + 1)) = 1, which is not zero.

• Important Note: If lim_n→∞ a_n = 0, the test is inconclusive. The series might converge or diverge; further testing is required.

2. Convergence Tests for Positive Term Series

These tests apply when all terms of the series are positive (or at least non-negative).

• Integral Test:

  • Description: If f(x) is a continuous, positive, and decreasing function for x ≥ 1, and f(n) = a_n, then the series ∑a_n and the integral ∫₁^∞ f(x) dx either both converge or both diverge.

  • Use Case: Effective for series where the corresponding integral is easy to evaluate.

• Comparison Test:

  • Description: Compare the given series to a known convergent or divergent series.

    • If 0 ≤ a_n ≤ b_n for all n, and ∑b_n converges, then ∑a_n converges.
    • If a_n ≥ b_n ≥ 0 for all n, and ∑b_n diverges, then ∑a_n diverges.

  • Use Case: Good for series that resemble well-known series like geometric or p-series.

• Limit Comparison Test:

  • Description: Let a_n > 0 and b_n > 0 for all n. If lim_n→∞ (a_n / b_n) = c, where 0 < c < ∞, then ∑a_n and ∑b_n either both converge or both diverge.

  • Use Case: Useful when direct comparison is difficult, but you can find a similar series for comparison.

• Ratio Test:

  • Description: Calculate L = lim_n→∞ |a_n+1 / a_n|.

    • If L < 1, the series converges absolutely.
    • If L > 1 (or L = ∞), the series diverges.
    • If L = 1, the test is inconclusive.

  • Use Case: Well-suited for series involving factorials or exponential terms.

• Root Test:

  • Description: Calculate L = lim_n→∞ ⁿ√|a_n|.

    • If L < 1, the series converges absolutely.
    • If L > 1 (or L = ∞), the series diverges.
    • If L = 1, the test is inconclusive.

  • Use Case: Effective for series where the entire term is raised to the power of n.

3. Convergence Tests for Alternating Series

These tests apply specifically to series where the terms alternate in sign.

• Alternating Series Test (Leibniz's Test):

  • Description: If the series is of the form ∑ (-1)ⁿ b_n or ∑ (-1)ⁿ⁺¹ b_n (where b_n > 0) and satisfies the following two conditions:

    1. b_n+1 ≤ b_n for all n (the terms are decreasing in magnitude).
    2. lim_n→∞ b_n = 0.

  • Then the alternating series converges.

  • Use Case: Specifically designed for alternating series.

4. Absolute vs. Conditional Convergence

• Absolute Convergence: If ∑ |a_n| converges, then ∑ a_n converges absolutely. Absolute convergence implies convergence.
• Conditional Convergence: If ∑ a_n converges, but ∑ |a_n| diverges, then ∑ a_n converges conditionally. This means the convergence depends on the alternating signs.

Summary Table of Convergence Tests

Test	Condition	Conclusion
Divergence Test	limn→∞ an ≠ 0	Diverges
Integral Test	f(x) continuous, positive, decreasing, f(n) = an	∑an converges if ∫1∞ f(x) dx converges; diverges if ∫1∞ f(x) dx diverges
Comparison Test	0 ≤ an ≤ bn, ∑bn converges OR an ≥ bn ≥ 0, ∑bn diverges	Converges if smaller than convergent; Diverges if larger than divergent
Limit Comparison Test	limn→∞ (an / bn) = c, 0 < c < ∞	Both converge or both diverge
Ratio Test	limn→∞	an+1 / an
Root Test	limn→∞ n√	an
Alternating Series Test	bn+1 ≤ bn, limn→∞ bn = 0	Converges

General Strategy

1. Check for Divergence: Always start with the divergence test.
2. Identify the Series Type: Is it a geometric series, p-series, alternating series, or something else?
3. Choose the Appropriate Test: Select the test that seems most likely to succeed based on the form of the series. If one test fails or is inconclusive, try another.
4. Be Mindful of Conditions: Ensure the conditions for the test you choose are met.

By systematically applying these tests, you can determine whether an infinite sum converges or diverges.