An infinite series is the sum of infinitely many terms, one after another forever. Remarkably, some such infinite sums add up to a specific, finite value (they "converge"), while others grow without bound or oscillate forever (they "diverge"). Determining which is which — and precisely what a converging infinite sum even means — is one of the foundational achievements of mathematical analysis.
Add ½ + ¼ + ⅛ + 1/16 + ... forever, doubling the denominator each time. You're adding infinitely many positive numbers together. Common sense might suggest the total should grow forever too — but it doesn't. This particular infinite sum adds up to exactly 1, no more and no less. This is a convergent series.
Recall the ancient Greek puzzle: to cross a room, you must first go halfway, then half the remaining distance, then half of that, forever — infinitely many steps. The resolution: these infinitely many distances (½ + ¼ + ⅛ + ...) add up to exactly the finite distance across the room. Infinite series are precisely the mathematical tool that makes rigorous sense of "infinitely many steps adding up to something finite."
A series Σaₙ converges to a sum S if the sequence of partial sums Sₙ = a₁+a₂+...+aₙ converges to S as n→∞ (in the precise epsilon-delta sense from the Limit entry). If the partial sums don't approach any finite value, the series diverges.
The ½+¼+⅛+... example above is the case a=½, r=½, giving sum = (½)/(1−½) = 1.
The series 1 + ½ + ⅓ + ¼ + ⅕ + ... (the harmonic series) diverges — even though its individual terms shrink toward zero, they don't shrink fast enough, and the sum grows (very slowly) without bound. This was first rigorously proven by Nicole Oresme in the 14th century, using a clever grouping argument, well before the formal machinery of calculus existed.
A series converges absolutely if Σ|aₙ| converges (a strong form of convergence). A series converges conditionally if it converges, but Σ|aₙ| diverges — meaning the convergence depends delicately on the specific alternating pattern of positive and negative terms. The alternating harmonic series 1−½+⅓−¼+... converges conditionally (to ln 2), even though the ordinary harmonic series without alternating signs diverges.
For a conditionally convergent series, the specific order in which you add the terms actually matters — a genuinely surprising and initially disturbing fact for anyone used to ordinary finite addition (where order never matters). Riemann proved that any conditionally convergent series of real numbers can be rearranged (by simply reordering, never adding or removing any terms) to converge to any target real number whatsoever, or even to diverge — a vivid illustration of why doing "safe," ordinary-arithmetic-like operations on infinite series requires considerably more mathematical care than one might initially assume.
A power series Σaₙxⁿ is a series involving powers of a variable x — for example, the Taylor series for eˣ, sin(x), and cos(x) (see the Derivative entry) are all power series. Every power series has a radius of convergence R such that the series converges absolutely for |x|<R and diverges for |x|>R (behaviour exactly at |x|=R requires separate, case-by-case analysis). This concept is essential to understanding exactly where a given Taylor series expansion is a mathematically valid representation of its original function.
Euler (1734) famously solved a problem that had stumped mathematicians for nearly a century: find the exact sum of the reciprocals of the perfect squares, 1 + 1/4 + 1/9 + 1/16 + .... Euler's surprising and beautiful answer: the sum equals exactly π²/6 — an unexpected and striking appearance of π in a problem that, on its surface, seems to have nothing at all to do with circles or geometry. This result substantially established Euler's early mathematical reputation and remains one of the most celebrated results in classical analysis.
The Basel Problem is the special case s=2 of the Riemann zeta function, ζ(s) = Σ1/nˢ (see the Riemann Hypothesis entry). Euler also found exact closed-form values for ζ(s) at all positive even integers, but the values at odd integers (like ζ(3), known as Apéry's constant) remain far more mysterious — Roger Apéry proved in 1978 that ζ(3) is irrational, a genuinely difficult and celebrated result, but whether ζ(3) can be expressed in any comparably simple closed form remains unknown even today.
Remarkably, certain formally divergent series (which do not converge in the ordinary sense at all) can still be practically useful in physics as asymptotic series — providing excellent numerical approximations when truncated after just the right number of terms, even though adding further additional terms eventually makes the approximation worse rather than better. This subtle and initially quite counterintuitive phenomenon appears throughout quantum field theory and perturbation theory in theoretical physics, and represented a significant conceptual challenge for 19th and 20th century mathematicians and physicists in properly and rigorously understanding what such divergent-yet-useful series actually meant.