Series Convergence Calculator

A common mistake is assuming that if the terms of a series approach zero, the series must converge. The harmonic series ∑1/n proves that is false: its terms go to zero, but the series diverges. The series convergence calculator runs a battery of convergence tests automatically, so you don't have to guess which test applies or whether you applied it correctly. It tells you whether the series converges, which test proves it, and why.

You enter the general term aₙ, pick a starting index, and choose a preset or type your own expression. The calculator then applies the divergence test, geometric series test, p-series test, ratio test, root test, limit comparison test, condensation test, and alternating series test in turn. For series whose form is exact, geometric, p-series, and certain logarithmic powers, the test is exact. For all others, the calculator estimates limits by evaluating aₙ at values up to n = 10⁸ and reports that the result is based on numeric evidence, not a proof. When a known closed form exists, the calculator shows the sum.

The calculator is for calculus students who need to verify homework, for AP Calculus BC students preparing for the exam, and for tutors checking student work. It is not for finding the exact sum of a convergent series; for that you need a resource on telescoping sums, geometric series formulas, or Fourier series.

Select a series form from the dropdown: General Term (Σ aₙ) or Function (Σ f(n)). Then choose a term type: Rational (p(n)/q(n)), Exponential (aⁿ), Factorial (n!), Power (nᵖ), Logarithmic (ln(n)), Mixed Expression, or Alternating Series. For each type, the interface shows the relevant input fields, numerator, denominator, base, multiplier, coefficient, exponent, or sign pattern.

Preset Series for Quick Testing

Eight presets let you jump straight to a well-known series: Harmonic Series (∑1/n), p-Series p=2 (∑1/n²), p-Series p=½ (∑1/√n), Geometric r=½ (∑(½)ⁿ), Geometric r=2 (∑2ⁿ), Alternating Harmonic (∑(−1)ⁿ/n), Exponential (∑1/n!), and Leibniz π/4 (∑(−1)ⁿ/(2n+1)). Selecting a preset fills in the expression and starting index automatically.

Starting Index

Enter the value of n at which the series starts. For ∑1/n, start at n=1. For ∑1/n!, start at n=0. The calculator uses this index to compute the first terms and partial sums.

Custom Expressions

Type any valid expression using n, the operations listed above, and parentheses. Implicit multiplication works: 2n means 2·n, and n(n+1) means n·(n+1). The convergence test calculator parses the expression and compiles it into a function it can evaluate for any n.

The calculator outputs one of four results: Converges Absolutely means Σ |aₙ| converges. Converges Conditionally means Σ aₙ converges but Σ |aₙ| diverges, this only happens for alternating series. Diverges means the partial sums do not approach a finite limit. Inconclusive means the tests the calculator tried could not decide; try a different test manually.

If the series converges absolutely and a closed-form sum is known, the calculator shows it. For example, ∑(½)ⁿ (n≥0) shows a sum of 2. For conditional convergence, the sum is shown if known, ∑(−1)ⁿ/n converges to −ln 2, not +ln 2 (a common sign error corrected here).

For numeric tests (ratio, root, limit comparison, integral, condensation), the calculator warns: This test estimated the limit from values up to n = 10⁸. That is strong evidence, not a proof. For exact tests (geometric, p-series, integral on power-log forms), it states the test is exact.

In automatic mode, the calculator runs the divergence test first. If lim aₙ ≠ 0, the series diverges and no further tests are needed. If the limit appears to be zero, the calculator checks the form of aₙ for a geometric pattern (constant ratio r) or a power pattern (constant exponent p). If it finds geometric form, it applies the geometric series test. If it finds p-series form, it applies the p-series test.

If the form is not exact, the calculator tries the ratio test (best for factorials and exponentials), then the root test (good for nth powers), then the limit comparison test (compares to a p-series or geometric series), then the condensation test (for series with logarithms), and finally the alternating series test if the terms alternate sign. The first test that gives a decisive result stops the process.

Why One Test Fails and Another Succeeds

The ratio test is inconclusive (L=1) for p-series and many rational series. For ∑1/n², the ratio test gives L=1, telling you nothing. The p-series test (p=2 > 1) gives a decisive 'converges'. Conversely, the p-series test cannot handle ∑n!/nⁿ, but the ratio test gives L=1/e < 1, so it converges. The calculator's automatic mode exploits these strengths: it matches the test to the series shape.

Each preset demonstrates a different convergence pattern. Run these examples to see how the calculator interprets the series and which test it selects.

Example 1: Geometric Series (r=½), Converges Absolutely

Preset: Geometric (r=½), Σ(½)ⁿ, n≥1. The calculator detects a geometric form with r=½. The geometric series test is exact. Since |½| < 1, the series converges absolutely. The sum is a/(1−r) = (½)/(1−½) = 1. This is the classic example of a convergent series used in the does the series converge calculator test.

Example 2: Harmonic Series (p=1), Diverges

Preset: Harmonic Series, Σ1/n, n≥1. The calculator detects a p-series form with p=1. The p-series test is exact. Since p ≤ 1, the series diverges. This is the standard counterexample: terms go to zero but the series diverges, a key fact any infinite series calculator must handle correctly.

Example 3: Alternating Harmonic, Converges Conditionally

Preset: Alternating Harmonic, Σ(−1)ⁿ/n, n≥1. The terms alternate sign, and bₙ = 1/n decreases to 0. The alternating series test gives conditional convergence. The sum is −ln 2, not +ln 2. This is a common error: the series starting at n=1 is −ln 2, because the first term is −1. A convergent or divergent calculator that gets the sign wrong is incorrect.

Example 4: p-Series (p=2), Converges Absolutely

Preset: p-Series (p=2), Σ1/n², n≥1. The p-series test with p=2 > 1 gives absolute convergence. The sum is π²/6 (the Basel problem, solved by Euler in 1734). The calculator shows the closed form if known.

Example 5: Exponential (∑1/n!), Converges Absolutely

Preset: Exponential, Σ1/n!, n≥0. This series is not geometric (no constant ratio) and not a p-series (the terms decay faster than any power). The ratio test gives L = lim |1/(n+1)| = 0 < 1, so the series converges absolutely. The sum is e. Note: a previous version of the preset evaluated n!/1 (which diverges) instead of 1/n!; this calculator uses the correct expression 1/n!.

Example 6: Leibniz Series, Converges Conditionally

Preset: Leibniz (π/4), Σ(−1)ⁿ/(2n+1), n≥0. This alternating series has bₙ = 1/(2n+1), which decreases to 0. The alternating series test gives conditional convergence. The sum is π/4, a classic result for the convergence test calculator to verify.

For most series that are not geometric, p-series, or simple power-log forms, the calculator estimates limits from values of aₙ up to n = 10⁸. That is strong evidence, but it is not a proof. A series that converges extremely slowly, like ∑1/n^(1.0001), may appear to converge from the first 10⁸ terms but actually converge (it is a p‑series with p = 1.0001 > 1, so it converges, albeit very slowly). The infinite series calculator labels such results as 'estimated', and you should treat them as a strong hint, not a guarantee. The single thing that most often goes wrong is trusting a numeric test for a series on the borderline of convergence (p near 1, ratio near 1) without also checking a different test manually.