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Positive Series

Positive Series

A series with terms that are all positive.

 

See also

Integral test, integral test remainder

Key Formula

∑n=1∞anwhere an>0 for all n\sum_{n=1}^{\infty} a_n \quad \text{where } a_n > 0 \text{ for all } n
Where:
  • ana_n = The nth term of the series, which must be strictly positive
  • nn = The index of summation, typically starting at 1

Worked Example

Problem: Determine whether the series is a positive series and whether it converges: the sum from n = 1 to infinity of 1/n².
Step 1: Check that every term is positive. For all positive integers n, n² > 0, so 1/n² > 0.
an=1n2>0for all n≥1a_n = \frac{1}{n^2} > 0 \quad \text{for all } n \geq 1
Step 2: Since every term is positive, this is indeed a positive series. The partial sums are increasing: S₁ = 1, S₂ = 1.25, S₃ ≈ 1.361, and so on.
S1=1,S2=1+14=54,S3=54+19=4936S_1 = 1, \quad S_2 = 1 + \frac{1}{4} = \frac{5}{4}, \quad S_3 = \frac{5}{4} + \frac{1}{9} = \frac{49}{36}
Step 3: Because this is a positive series, we can apply the Integral Test. The function f(x) = 1/x² is positive, continuous, and decreasing on [1, ∞). Evaluate the improper integral.
∫1∞1x2 dx=[−1x]1∞=0−(−1)=1\int_1^{\infty} \frac{1}{x^2}\,dx = \left[-\frac{1}{x}\right]_1^{\infty} = 0 - (-1) = 1
Step 4: The integral converges (to 1), so by the Integral Test the series also converges. In fact, its exact sum is known to be π²/6.
∑n=1∞1n2=π26≈1.6449\sum_{n=1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6} \approx 1.6449
Answer: The series ∑ 1/n² is a positive series, and it converges to π²/6.

Another Example

Problem: Is the series ∑ from n = 1 to infinity of 1/n (the harmonic series) a positive series? Does it converge?
Step 1: Each term is 1/n, which is positive for all n ≥ 1, so this is a positive series.
an=1n>0a_n = \frac{1}{n} > 0
Step 2: Apply the Integral Test. The integral of 1/x from 1 to infinity diverges.
∫1∞1x dx=lim⁡b→∞ln⁡b=∞\int_1^{\infty} \frac{1}{x}\,dx = \lim_{b \to \infty} \ln b = \infty
Step 3: The integral diverges, so the harmonic series diverges. Being a positive series does not guarantee convergence—it only guarantees the partial sums increase without oscillating.
Answer: The harmonic series is a positive series, but it diverges to infinity.

Frequently Asked Questions

Why do positive series matter for convergence tests?
Many convergence tests—such as the Integral Test, Comparison Test, Limit Comparison Test, and Ratio Test—require (or are most naturally stated for) positive series. The key reason is that when all terms are positive, the partial sums form a monotone increasing sequence. By the Monotone Convergence Theorem, such a sequence converges if and only if it is bounded above, which makes the convergence question simpler to analyze.
Is a positive series the same as a nonnegative series?
Not exactly. A positive series requires every term to be strictly greater than zero (aₙ > 0). A nonnegative series allows terms to equal zero (aₙ ≥ 0). In practice, having finitely many zero terms does not affect convergence, so the distinction rarely matters for convergence tests. However, the standard definition of a positive series uses strict positivity.

Positive Series vs. Alternating Series

A positive series has all terms positive, so its partial sums always increase. An alternating series has terms that alternate in sign (positive, negative, positive, …), so its partial sums bounce above and below the eventual limit. Different convergence tests apply: positive series use the Integral Test, Comparison Test, or Ratio Test, while alternating series use the Alternating Series Test. A series can converge as an alternating series even when the corresponding positive series (its absolute values) diverges—this is called conditional convergence.

Why It Matters

Identifying a series as positive unlocks the most commonly used convergence tests in calculus, including the Integral Test, Direct Comparison Test, Limit Comparison Test, and Ratio Test. The monotone-increasing behavior of the partial sums simplifies analysis considerably: you only need to show the sums are bounded to prove convergence. Positive series also form the foundation for understanding absolute convergence, since testing whether ∑∣an∣\sum |a_n| converges means testing a positive series.

Common Mistakes

Mistake: Assuming a positive series must converge because all terms are positive and getting smaller.
Correction: Decreasing positive terms are necessary but not sufficient for convergence. The harmonic series ∑ 1/n has decreasing positive terms but diverges. You must apply an actual convergence test.
Mistake: Applying the Comparison Test or Integral Test to a series with some negative terms without taking absolute values first.
Correction: These tests are designed for positive series. If your series has negative terms, either take absolute values (to test for absolute convergence) or use a test suited for series with mixed signs, such as the Alternating Series Test.

Related Terms