Converge or Diverge?
An infinite series Σaₙ converges if its partial sums settle toward a finite number and diverges if they do not. The first question is always whether the terms go to zero: if aₙ does not tend to 0, the series diverges, full stop. The converse is false, and the harmonic series shows it.
Σ1/n has terms that go to 0, yet its sum has no limit. After one million terms the partial sum is about 14.39, which is ln(10⁶) + 0.5772. To pass 100 you would need around 1.5 × 10⁴³ terms, but it still never stops growing.
Which Test Should You Use?
| If the term has | Try | Typical verdict |
|---|---|---|
| A constant ratio, rⁿ | Geometric | Converges if |r| < 1 |
| 1/nᵖ | p-series | Converges if p > 1 |
| Factorials or exponentials, n!, 2ⁿ | Ratio test | L < 1 converges, L > 1 diverges |
| An nth power, (…)ⁿ | Root test | Same cut-offs as ratio |
| Looks like a known series | Comparison, limit comparison | Match growth rates |
| f(n) positive and decreasing | Integral test | Same fate as ∫f |
| (−1)ⁿ times a decreasing term | Alternating series test | Converges if terms → 0 |
The ratio and root tests say nothing when L = 1. That is exactly where p-series live, so use comparison or the integral test.
Worked Examples
Σ 3ⁿ ÷ n!
- Ratio: an+1/aₙ = 3 ÷ (n + 1).
- The limit as n → ∞ is 0, and 0 < 1.
Answer: Converges (the sum from n = 0 is e³ ≈ 20.09)
Σ 1/n² from n = 1
- This is a p-series with p = 2 > 1.
- Euler found the exact sum in 1735.
Answer: Converges to π²/6 ≈ 1.6449
Σ (−1)ⁿ⁺¹/n
- Terms 1/n decrease to 0, so the alternating series test gives convergence.
- Absolute values give the harmonic series, which diverges.
Answer: Converges conditionally to ln 2 ≈ 0.6931
Σ 1/(n ln n) vs Σ 1/(n (ln n)²)
- Integral test with u = ln x: ∫dx/(x ln x) = ln ln x → ∞.
- ∫dx/(x (ln x)²) = −1/ln x converges.
Answer: The first diverges, the second converges
Power Series: Radius and Interval of Convergence
A power series Σcₙ(x − a)ⁿ converges for x close enough to a. The ratio test gives the radius R, and you must test the two endpoints separately because the test is silent there.
| Series | Radius | Interval |
|---|---|---|
| Σ xⁿ/n | 1 | [−1, 1) |
| Σ xⁿ/n² | 1 | [−1, 1] |
| Σ xⁿ/n! | ∞ | (−∞, ∞) |
| Σ (x−3)ⁿ/2ⁿ | 2 | (1, 5) |
Absolute vs Conditional Convergence
If Σ|aₙ| converges the series converges absolutely, and you can rearrange the terms freely. If Σaₙ converges but Σ|aₙ| does not, it converges conditionally. Riemann showed that a conditionally convergent series can be rearranged to add up to any number you like, which is why the distinction matters.
What We Think Is Overrated, Underrated and Wrong
Original Data: A Divergence You Cannot See
The harmonic series 1 + 1/2 + 1/3 + … diverges, but painfully slowly: after 1,000,000 terms the sum is only about 14.39 (ln N + 0.5772). By contrast Σ1/n² converges to π²/6 = 1.6449. Numbers alone cannot distinguish these cases, which is why you need a test.
Series You Are Asked About
| Series | Result | Test |
|---|---|---|
| Σ 1/nᵖ | converges if p > 1 | p-series |
| Σ rⁿ | converges if |r| < 1, sum 1 ÷ (1 − r) | geometric |
| Σ (−1)ⁿ⁺¹/n | converges to ln 2 (conditionally) | alternating series |
| Σ n!/nⁿ | converges | ratio test (limit 1/e) |
Our Take: What Is Overrated, Underrated and Plain Wrong
Overrated: the ratio test for everything. It fails on p-series, where the ratio tends to 1.
Underrated: the nth-term test as a first check. If the terms do not tend to 0, you are done: it diverges.
Plain wrong: believing that terms going to zero is enough. 1/n goes to zero and still diverges.
Series Mistakes That Cost Marks
- Concluding convergence from aₙ → 0.
- Using the ratio test when L = 1 and treating it as a verdict.
- Applying the alternating test when the terms do not decrease.
- Forgetting to test endpoints of a power series.
- Comparing with a series that goes the wrong way: bigger than a divergent one proves nothing.
Why Students Get Stuck on Series Convergence, and What a Good Calculator Should Do
Searching for help with series convergence usually starts late at night with one problem that will not work out. There are many tests, and the right one is not obvious. A good calculator fixes that by showing each step, checking its own answer and explaining the idea in plain words. This section covers what students struggle with, what they want, and how this page responds.
In short: students need to choose a test and justify the result, not just see a final number, and a calculator earns trust only when its steps can be checked.
Understand the pain points
What students get stuck on
Most mistakes here are not about arithmetic. They come from one rule applied the wrong way round, which is why seeing every step matters.
- Not knowing which test to try firstDoes the series sum of 1/n^2 converge?↗
- Concluding convergence from terms going to zeroDoes the series sum of 1/n converge?↗
- Misusing the ratio test when the limit is 1Does the series sum of (-1)^n / n converge?↗
- Mixing up convergence and absolute convergence
Understand the frustrations and desires
What they feel, and what they want
Each frustration points to a wish: a clearer step, a built-in check, an explanation in plain words.
- Wants a test suggested for the series
- Wants the limit step shown
- Wants the conclusion stated clearly
Build the page around their intent
Each intent has a place on this page
The page follows intent, not features, so each need has a clear starting point.
- I need an answer now: type the problem into the calculator at the top of this page.Does the series sum of 1/n^2 converge?↗
- I need to understand: the sections below explain the idea in plain words and list the common mistakes.
- I need to check my work: every step is shown, so the first line that differs from yours is the one to study.
- I need practice: open the solver, tap + and choose Create Practice Test.
- I need it explained aloud: in the solver, tap + and choose Create Video.