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Alternating Series Test Calculator

Enter the exponent p to test whether the alternating p-series converges.

How it is calculated

The tool studies the alternating p-series Σ (−1)^(k+1) / k^p = 1 − 1/2^p + 1/3^p − …. The verdict depends only on the exponent p. If p ≤ 0 the terms do not shrink to zero, so the series diverges. If 0 < p ≤ 1 the terms decrease to zero, so the Leibniz test gives convergence, but the series of absolute values Σ1/k^p diverges, which makes it conditional. If p > 1 the absolute series converges too, so convergence is absolute. It also adds up the first N terms directly, with signs +, −, +, … starting from k = 1, and shows the partial sum to six decimals. N defaults to 100, is rounded to a whole number and is at least 1. The verdict covers the whole series, but the partial sum covers only N terms.

Worked example

Enter p = 1 and N = 100, which is the alternating harmonic series 1 − 1/2 + 1/3 − 1/4 + …. Since 0 < p ≤ 1, the verdict is conditional convergence: the terms fall toward zero, but the plain harmonic series diverges. Adding the first 100 terms gives 0.688172 (first 100 terms). The Leibniz error bound says the leftover is smaller than the first omitted term, 1/101 ≈ 0.0099, so the true sum lies within that distance of the partial sum. For p = 2 and N = 10 the same method gives 0.817962 with an absolute-convergence verdict.

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Common mistakes

  • Reading the partial sum as the limit. It is only the sum of the first N terms; the series may need many more to settle.
  • Confusing conditional and absolute convergence. For p between 0 and 1 the signs are what make the series converge; remove them and it diverges.
  • Applying the verdict to a different series. The tool only tests terms of the form 1/k^p, not other alternating series such as (−1)^k / ln k.
  • Entering p as a negative number and expecting convergence. With p ≤ 0 the terms do not go to zero and the partial sum just oscillates or grows.
  • Thinking N changes the verdict. The verdict depends on p alone; N only changes the displayed partial sum.

About this calculator

The alternating series test (Leibniz test) is a convergence test from calculus for series whose terms alternate in sign, such as Σ(−1)ⁿ⁺¹/nᵖ. It states that if the absolute value of the terms, bₙ = 1/nᵖ here, decreases monotonically toward zero as n grows, then the alternating series converges — a much easier condition to check than the general convergence tests needed for series without alternating signs. A useful bonus is the error bound: truncating the series after N terms leaves a remainder no larger in absolute value than the first omitted term, b_(N+1).

This calculator checks that monotonic-decrease-to-zero condition for a chosen exponent p, reports whether the alternating p-series converges by the Leibniz test, and computes the partial sum of the first N terms so you can see how quickly it’s approaching its limit — standard material in an introductory real-analysis or calculus II course on series convergence.

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