Integral test for convergence

The integral test determines whether a series of positive terms converges by comparing its partial sums with the improper integral of a positive, decreasing function that generates those terms.

Improper integral: An integral defined over an unbounded interval or with an unbounded integrand, evaluated through limits. The test asks whether the corresponding integral over an unbounded interval is finite.
Series (mathematics): An infinite sum defined as the limit of its sequence of finite partial sums. The test decides convergence of this limit by relating partial sums to an integral.
Logarithmic series: A family of series whose terms involve powers of n and logarithms of n. The test evaluates many such series by integrating their corresponding logarithmic functions.
Limit comparison test: A series test that compares the ratio of corresponding terms with a known series. It compares series directly and does not require constructing a decreasing function.
Leonhard Euler: An eighteenth-century mathematician whose work shaped analysis, number theory, and mathematical notation. Euler studied infinite series, including the harmonic series whose divergence the test makes transparent.
Monotone function: A function that either never decreases or never increases across its domain. The function must decrease so its graph bounds the series terms by adjacent rectangles.
Convergence of a series: The property that a sequence of partial sums approaches a finite limit. This is the behavior the integral test is designed to classify.
Generalized harmonic number: The finite sum of the first n reciprocal powers, often written as the nth partial sum of a p-series. Integral bounds estimate these partial sums and explain their growth.
Ratio test: A convergence test based on the limit of the ratio of consecutive absolute terms. It often handles factorial or exponential terms more effectively than an integral comparison.
Augustin-Louis Cauchy: A nineteenth-century mathematician who helped establish rigorous foundations for analysis. His program of rigor made precise the convergence questions addressed by series tests.

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