P-series convergence checker

Enter any finite real exponent, and the calculator compares it with the critical value 1. The response states whether the series converges, shows the relevant comparison, and repeats the governing criterion. It is a direct, deterministic way to check homework, document a calculation, or add the standard p-series test to an automated workflow without estimating partial sums.

Recognize the p-series before applying the test

The p-series test applies to an infinite sum whose nth term is exactly 1/n^p, starting at a positive integer index. The exponent p must be a fixed real number; it does not change with n. Once the series has this form, its behavior depends entirely on whether p lies above the critical threshold 1. Enter that exponent in the p field. For example, sum 1/n^2 has p = 2, while the harmonic series sum 1/n has p = 1. A constant multiplier in front of the entire series does not change convergence when that constant is finite and nonzero, although this checker expects only the exponent because the multiplier is irrelevant to the test. If the terms contain logarithms, factorials, alternating signs, changing exponents, or a more complicated denominator, first use algebra or a comparison theorem to justify reducing the question to a p-series. Do not extract a convenient-looking exponent from an expression that is not actually equivalent to this standard form.

Compare p with the critical value one

A p-series converges if and only if p is strictly greater than 1. The word strictly matters. When p = 1, the series is the harmonic series and diverges, even though its terms approach zero. When p is below 1, the series also diverges. This includes positive fractional exponents such as 1/2, zero, and negative exponents. At p = 0 every term equals 1, while a negative p makes the term magnitudes grow rather than shrink. For p > 1, the terms decay quickly enough for the infinite sum to approach a finite value. The checker returns the supplied exponent, a Boolean convergence result, a plain-language classification, and one of the comparisons p > 1, p = 1, or p < 1. The threshold field remains 1 so software can consume the result without parsing prose. No numerical tolerance is used: the input number is compared directly with the exact mathematical boundary represented by 1.

Interpret the result and its limits

A convergent classification says that the sequence of partial sums approaches a finite limit. It does not generally calculate that limit. Some p-series have familiar values, such as p = 2, but the threshold test itself only decides convergence. A divergent classification says that no finite sum is approached under the ordinary definition of an infinite series. It does not mean every term grows, nor does it mean the first several partial sums must look large; divergence at p = 1 is famously slow. Also remember that terms approaching zero are necessary but not sufficient for convergence. This capability is therefore best used as a theorem-based decision, not as a numerical simulation. When auditing a comparison-test solution, verify the inequality direction separately: comparison with a convergent p-series can establish convergence under an upper bound, while comparison with a divergent p-series can establish divergence under a lower bound. The checker classifies the reference p-series, but it cannot verify an unstated inequality involving another sequence. Each API calculation costs $0.002, and the browser version can apply the same deterministic rule locally.