How to Determine Series Convergence
Determining whether an infinite series converges or diverges is a central challenge in calculus. An infinite series is the sum of an infinite sequence of terms. If the partial sums approach a finite limit, the series converges; otherwise, it diverges. This guide covers the essential convergence tests.
The divergence test is the first check: if the limit of a_n as n approaches infinity is not zero, the series diverges. This is a necessary but not sufficient condition. If the limit is zero, the series may converge or diverge, and you need further tests.
The geometric series sum ar^n converges when the absolute value of r is less than one, with sum a over (1 minus r). This is the most important series to recognize because many other tests compare unknown series to geometric series. Our series convergence tester lets you explore geometric, p-series, alternating, and factorial series.
The p-series sum 1/n^p converges when p exceeds one and diverges when p is at most one. The case p equals one gives the harmonic series, one of the most counterintuitive results in mathematics: even though the terms approach zero, the sum diverges to infinity.
The integral test compares a series to an improper integral. If f(n) equals a_n and f is positive, continuous, and decreasing, then the series and the integral either both converge or both diverge. This test is especially useful for series involving logarithms.
The comparison and limit comparison tests compare a series to a known benchmark. If zero is less than or equal to a_n less than or equal to b_n and sum b_n converges, then sum a_n converges. If a_n is greater than or equal to c_n greater than zero and sum c_n diverges, then sum a_n diverges.
The ratio test examines the limit of a_(n+1)/a_n. If the limit is less than one, the series converges absolutely. If greater than one, it diverges. If equal to one, the test is inconclusive. The root test examines the nth root of the absolute value of a_n and follows the same classification.
The alternating series test applies to series of the form sum (-1)^(n-1) b_n where b_n is positive, decreasing, and approaches zero. Such series always converge. The error from truncation is bounded by the first omitted term. Our summation calculator computes partial sums to estimate convergence behavior.