succesiune

Succesiune, in mathematics, refers to an ordered list of elements indexed by natural numbers. It is commonly represented as a_n or as (a_n)_{n∈N}, where a_n denotes the nth term. The terms may come from any set, with real numbers being the most common case. The concept focuses on the arrangement and behavior of the terms rather than on a single sum.

Common examples include arithmetic succesiune, defined by a_n = a + (n−1)d, and geometric succesiune, defined by a_n = a r^{n−1}. The Fibonacci sequence is a well-known example, defined by F_1 = 1, F_2 = 1, and F_n = F_{n−1} + F_{n−2} for n > 2.

Convergence and limits are central to the study of succesiuni. A sequence (a_n) of real numbers converges to a limit L if, for every ε > 0, there exists N such that n ≥ N implies |a_n − L| < ε. If no such L exists, the sequence diverges. Many real-valued sequences that are bounded and monotone converge, a principle used throughout analysis.

Other important notions include monotonicity (increasing or decreasing), boundedness, and the concept of Cauchy sequences, which capture the idea that terms become arbitrarily close to each other as the index grows. A subsequence is obtained by selecting terms with strictly increasing indices; subsequences can have limits different from the original sequence. The limsup and liminf describe the ultimate upper and lower bounds of a sequence’s tails.

Relation to series: a sequence lists terms, while a series sums them (the partial sums form a sequence themselves). Succesiune is foundational in calculus, analysis, and numerical methods, and is widely used across mathematical literature to describe ordered progressions of terms.