- Related Topics:
- zeta function
- Dirichlet’s test
- partial sum
- On the Web:
- Story of Mathematics - Infinite series (Nov. 02, 2026)
infinite series
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What is an infinite series in mathematics?
An infinite series is the sum of an endless sequence of numbers that follow a specific mathematical pattern. For example, the series 1/2 + 1/4 + 1/8 + ... continues indefinitely.
In mathematics, we often look at "partial sums," which are sums of a finite number of terms from the series. If these partial sums approach a specific, fixed number as more terms are added, the series is said to converge, and that number is the sum of the series. If the partial sums do not approach a fixed number, the series diverges. Infinite series are valuable tools in various scientific and engineering fields.
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How is an infinite series different from a finite sum?
An infinite series involves the sum of an endless sequence of numbers, while a finite sum involves adding a limited number of terms. The key difference lies in whether the summation continues indefinitely or stops after a specific term.
An infinite series is defined as the sum of infinitely many numbers, such as a1 + a2 + a3 + .... A finite sum, also known as a partial sum, is calculated by adding only the first n terms of a series: sn = a1 + a2 + ... + an.
If the partial sums of an infinite series approach a fixed number S as n increases, the series is said to converge, and S is its sum. If the partial sums do not approach a finite value, the series diverges, and no sum is assigned. For example, the infinite series 1 + 1 + 1 + ... diverges because its partial sums grow without bound. In contrast, the series 1/2 + 1/4 + 1/8 + ... converges to 1.
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What does it mean for an infinite series to converge?
For an infinite series to converge, its partial sums must approach a fixed number as more terms are added. This fixed number is called the sum of the series. If the partial sums do not approach a specific value, the series is said to diverge. For example, the series 1 + 1/2 + 1/4 + 1/8 + ... converges to 1 because its partial sums get progressively closer to 1. Conversely, the series 1 + 1 + 1 + ... diverges because its partial sums grow indefinitely without approaching any finite number.
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Can you give an example of a convergent infinite series?
A convergent infinite series is one where the sum of its terms approaches a fixed finite value as more terms are added. A classic example is the geometric series:
1 + 1/2 + 1/4 + 1/8 + ...
This series, where each term is half of the previous one, converges to a sum of 2. The partial sums get progressively closer to 2. In contrast, a series like 1 + 1 + 1 + ... diverges because its partial sums grow infinitely large. The study of such series is fundamental to mathematics and has applications across various scientific fields.
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What is a divergent infinite series?
An infinite series is the sum of an endless sequence of numbers that follow a specific pattern. When the partial sums of an infinite series do not approach a fixed finite value as more terms are added, the series is said to diverge. In such cases, no sum is assigned to the series. For instance, the series 1 + 1 + 1 + ... diverges because its partial sums grow without bound. Another example is the series 1 - 1 + 1 - 1 + ..., which diverges because its partial sums oscillate between 0 and 1 and never settle on a single value. A series that does not converge is a divergent series.
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How do mathematicians test if an infinite series converges or diverges?
Mathematicians determine if an infinite series converges or diverges by examining its partial sums. If the partial sums approach a fixed number as more terms are added, the series converges to that number. If the partial sums do not approach a finite value, the series diverges.
Several tests can help determine convergence or divergence:
* Comparison Test: If a series' terms are less than or equal to the terms of a known convergent series, then the series also converges.
* Ratio Test: For series with positive terms, if the ratio of consecutive terms is less than 1, the series converges.
* Dirichlet's Test: This test applies when a series' partial sums are bounded and a sequence of terms monotonically decreases to zero.
* Weierstrass M-test and Abel's test are also used, particularly for uniform convergence.If a series converges, its individual terms must approach zero. However, terms approaching zero do not guarantee convergence.
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infinite series, the sum of infinitely many numbers related in a given way and listed in a given order. Infinite series are useful in mathematics and in such disciplines as physics, chemistry, biology, and engineering.
For an infinite series a1 + a2 + a3 +⋯, a quantity sn = a1 + a2 +⋯+ an, which involves adding only the first n terms, is called a partial sum of the series. If sn approaches a fixed number S as n becomes larger and larger, the series is said to converge. In this case, S is called the sum of the series. An infinite series that does not converge is said to diverge. In the case of divergence, no value of a sum is assigned. For example, the nth partial sum of the infinite series 1 + 1 + 1 +⋯ is n. As more terms are added, the partial sum fails to approach any finite value (it grows without bound). Thus, the series diverges. An example of a convergent series is 
As n becomes larger, the partial sum approaches 2, which is the sum of this infinite series. In fact, the series 1 + r + r2 + r3 +⋯ (in the example above r equals 1/2) converges to the sum 1/(1 − r) if 0 < r < 1 and diverges if r ≥ 1. This series is called the geometric series with ratio r and was one of the first infinite series to be studied. Its solution goes back to Zeno of Elea’s paradox involving a race between Achilles and a tortoise (see mathematics, foundations of: Being versus becoming).
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Certain standard tests can be applied to determine the convergence or divergence of a given series, but such a determination is not always possible. In general, if the series a1 + a2 +⋯ converges, then it must be true that an approaches 0 as n becomes larger. Furthermore, adding or deleting a finite number of terms from a series never affects whether or not the series converges. Furthermore, if all the terms in a series are positive, its partial sums will increase, either approaching a finite quantity (converging) or growing without bound (diverging). This observation leads to what is called the comparison test: if 0 ≤ an ≤ bn for all n and if b1 + b2 +⋯ is a convergent infinite series, then a1 + a2 +⋯ also converges. When the comparison test is applied to a geometric series, it is reformulated slightly and called the ratio test: if an > 0 and if an + 1/an ≤ r for some r < 1 for every n, then a1 + a2 +⋯ converges. For example, the ratio test proves the convergence of the series 
Many mathematical problems that involve a complicated function can be solved directly and easily when the function can be expressed as an infinite series involving trigonometric functions (sine and cosine). The process of breaking up a rather arbitrary function into an infinite trigonometric series is called Fourier analysis or harmonic analysis and has numerous applications in the study of various wave phenomena.