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 s_n=a₁+a₂+⋯+ a_n, which involves adding only the first n terms, is called a partial sum of the series. If s_n approaches a fixed number S as n becomes larger and larger, the series is said to converge (convergence). 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 (mathematics, foundations of)).

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 a_n 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≤a_n≤b_n 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 a_n\\>0 and if a_{n + 1}/a_n≤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.