Convergent series

In mathematics, a series is the sum of the terms within an infinite sequence of numbers. While a sequence is simply a list of numbers, a series is what happens when you add those numbers together. To understand this, mathematicians look at partial sums. A partial sum is the total you get after adding only the first few terms of the sequence. If you keep adding terms one by one, you create a new sequence made of these partial sums.

A series is called convergent if its sequence of partial sums tends toward a specific limit. This means that as you add more and more terms, the total gets closer and closer to a single, fixed number. This unique number is known as the sum of the series. We often use the same mathematical symbol to represent both the operation of adding the terms and the final result. If a series does not settle on a specific number, it is called a divergent series. This occurs when the sum does not approach a finite limit.

Mathematicians use several specific tests to determine if a series will converge or diverge. The comparison test involves comparing a series to another one that is already understood. If a series is smaller than a known convergent series, then it must also converge. Conversely, if a series is larger than a known divergent series, it must diverge. The limit comparison test is a similar method. It checks if two series behave the same way by looking at the limit of their ratio.

Other tests look at the relationship between individual terms. The ratio test examines the limit of the ratio between consecutive terms. If this limit, denoted as r, is less than one, the series is absolutely convergent. If r is greater than one, the series diverges. The root test also uses a limit, but it focuses on the nth root of the terms. If this limit is less than one, the series converges. Both the ratio and root tests are based on comparisons with geometric series. The root test is generally more applicable, though the limit can be difficult to calculate.

There are also tests for specific types of series. The alternating series test, or Leibniz criterion, applies to series where the signs of the terms switch between positive and negative. If the terms decrease in size and approach zero, the series converges. The integral test connects series to calculus. It compares a series to the integral of a corresponding function. If the integral converges, the series also converges. Other advanced methods include the Cauchy condensation test and Dirichlet's test.

Series can be classified by how strongly they converge. An absolutely convergent series is one that remains convergent even if all its terms become positive. Every absolutely convergent series is also convergent. However, some series are conditionally convergent. These series converge in their original form but would diverge if all terms were made positive. The Maclaurin series for the logarithm function is a famous example of conditional convergence.

Conditional convergence leads to some very strange mathematical properties. The Riemann series theorem states that if a series is conditionally convergent, you can rearrange its terms to change the result. By changing the order of addition, you can make the series converge to any value you choose. You can even make it diverge entirely. This shows that for these delicate series, the order of the numbers matters deeply. In contrast, absolutely convergent series are much more stable regardless of their order.

We can see these behaviors in many different number patterns. The reciprocals of positive integers form the harmonic series, which is divergent. However, if you alternate the signs of those same reciprocals, you get the alternating harmonic series, which converges. The reciprocals of prime numbers also produce a divergent series, suggesting the set of primes is large. In contrast, the reciprocals of powers of two create a convergent series, showing that powers of two are a "small" set. Other convergent examples include the reciprocals of square numbers, known as the Basel problem, and the reciprocals of factorials.