tekensreeks

A tekensreeks, in mathematical context, is a sequence that records only the sign of terms from another sequence or function. It typically takes values in the set {-1, 0, 1} and is often defined by s_n = sign(a_n), where sign is the sign function: s_n = -1 if a_n < 0, s_n = 0 if a_n = 0, and s_n = 1 if a_n > 0. The tekensreeks thus encodes whether each term is negative, zero, or positive, without preserving its magnitude.

Purpose and use. Tekensreeksen are used to study the distribution of signs in a sequence or series. They help analyze oscillation, sign patterns, and convergence behavior where the sign arrangement matters, such as alternating or non-monotone terms. In some cases, the sign sequence guides the application of convergence tests that depend on the arrangement of signs, or facilitates the construction of examples with prescribed sign patterns.

Relations and interpretations. The tekensreeks is closely related to the sign function and to binary or alternating sequences. It provides a compact representation of sign information and can be used in conjunction with magnitude data to examine overall behavior. For instance, if a_n = (-1)^n b_n with b_n > 0, then the tekensreeks is an alternating sequence (-1, 1, -1, 1, …). Conversely, if some a_n are zero, the sign sequence will include zeros to reflect that.

Examples. For a_n = (-1)^n/n, the tekensreeks is (-1, 1, -1, 1, …). For a_n = n^2 - 5, the tekensreeks begins with (1, 1, 1, 1, -1, -1, …). The concept is general and applies to any real-valued sequence.

See also. Sign function, alternating series, convergence tests, sign pattern, binary sequence.