Series Tests

1. Geometric Series Test

• Definition: A series of the form ∑n=0∞arn\sum_{n=0}^\infty ar^n.

• Convergence: Converges if ∣r∣<1|r| < 1. The sum is a1−r\frac{a}{1 - r}.

• Divergence: Diverges if ∣r∣≥1|r| \geq 1.

2. P-Series Test

• Definition: A series of the form ∑n=1∞1np\sum_{n=1}^\infty \frac{1}{n^p}.

• Convergence: Converges if p>1p > 1.

• Divergence: Diverges if p≤1p \leq 1.

3. Test for Divergence (Nth-Term Test)

• Definition: For ∑an\sum a_n, if lim⁡n→∞an≠0\lim_{n \to \infty} a_n \neq 0 or does not exist, the series diverges.

• Note: If lim⁡n→∞an=0\lim_{n \to \infty} a_n = 0, the series may converge or diverge (use other tests).

4. Integral Test

• Definition: For a positive, continuous, decreasing function f(n)f(n) such that an=f(n)a_n = f(n):

  • ∑an\sum a_n and ∫f(x)dx\int f(x)dx either both converge or both diverge.

5. Comparison Test

• Definition: Compare ∑an\sum a_n with a known series ∑bn\sum b_n:

  • If 0≤an≤bn0 \leq a_n \leq b_n and ∑bn\sum b_n converges, then ∑an\sum a_n converges.

  • If 0≤bn≤an0 \leq b_n \leq a_n and ∑bn\sum b_n diverges, then ∑an\sum a_n diverges.

6. Limit Comparison Test

• Definition: For ∑an\sum a_n and ∑bn\sum b_n, let lim⁡n→∞anbn=c\lim_{n \to \infty} \frac{a_n}{b_n} = c, where c>0c > 0:

  • If ∑bn\sum b_n converges, ∑an\sum a_n also converges.

  • If ∑bn\sum b_n diverges, ∑an\sum a_n also diverges.

7. Alternating Series Test (Leibniz Test)

• Definition: For an alternating series ∑(−1)nan\sum (-1)^n a_n, where an>0a_n > 0:

  • Converges if ana_n is decreasing and lim⁡n→∞an=0\lim_{n \to \infty} a_n = 0.

8. Ratio Test

• Definition: For ∑an\sum a_n, compute L=lim⁡n→∞∣an+1an∣L = \lim_{n \to \infty} \left|\frac{a_{n+1}}{a_n}\right|:

  • If L<1L < 1, the series converges absolutely.

  • If L>1L > 1 or L=∞L = \infty, the series diverges.

  • If L=1L = 1, the test is inconclusive.

9. Root Test

• Definition: For ∑an\sum a_n, compute L=lim⁡n→∞∣an∣nL = \lim_{n \to \infty} \sqrt[n]{|a_n|}:

  • If L<1L < 1, the series converges absolutely.

  • If L>1L > 1 or L=∞L = \infty, the series diverges.

  • If L=1L = 1, the test is inconclusive.

10. Absolute Convergence Test

• Definition: If ∑∣an∣\sum |a_n| converges, then ∑an\sum a_n converges absolutely.

• Note: Absolute convergence implies convergence, but not vice versa.

11. Conditional Convergence

• Definition: A series ∑an\sum a_n converges conditionally if ∑an\sum a_n converges, but ∑∣an∣\sum |a_n| diverges.

12. Telescoping Series Test

• Definition: For a series where most terms cancel out, such as ∑(bn−bn+1)\sum (b_n - b_{n+1}):

  • Converges if the remaining terms lim⁡n→∞bn\lim_{n \to \infty} b_n exist and are finite.