Convergence Tests

In mathematics, particularly in calculus, we often deal with infinite series. It's important to know whether an infinite series converges (has a finite sum) or diverges (doesn't have a finite sum). Several tests can help us determine this.

Nth Term Test

The simplest test is the nth term test. If the limit of the nth term of a series as n approaches infinity is not zero, the series diverges.

Geometric Series Test

A geometric series a + ar + ar^2 + ar^3 + ... converges if the absolute value of the common ratio r is less than 1 and diverges otherwise.

P-Series Test

A p-series is of the form 1/n^p. If p > 1, the series converges. If p <= 1, the series diverges.

Comparison Test

If every term in series A is less than or equal to the corresponding term in series B, and series B converges, then series A also converges. If every term in series A is greater than or equal to the corresponding term in series B, and series B diverges, then series A also diverges.

Integral Test

The integral test states that a series converges if the integral of the function from which the series is derived converges.

Ratio Test and Root Test

The ratio test compares the ratio of successive terms, while the root test compares the nth root of the nth term, both as n approaches infinity. These tests can provide a measure of how fast the terms of the series are growing or shrinking.

Alternating Series Test

An alternating series is one in which the signs of the terms alternate between positive and negative. The alternating series test can be used to determine if such a series converges.

Absolute Convergence Test

If the series converges when all terms are replaced by their absolute values, then the series is said to converge absolutely. A series that converges absolutely will also converge when the absolute values are removed.

These are just a few of the most common convergence tests. Applying the correct test can often simplify the process of determining whether a series converges or diverges.