Limit of an Infinite Series in Single-Variable Calculus
An infinite series is defined as the limit of its sequence of partial sums: the series converges to a value L if and only if this limit exists, and diverges otherwise. This principle belongs to the theory of infinite series within single-variable calculus, a subfield of real analysis concerned with the behavior of sums of infinitely many terms. Establishing a closed-form expression for the partial sums (often via algebraic manipulation such as partial-fraction decomposition and confirmed by mathematical induction) allows the limit, and hence the sum of the series, to be determined directly.
Limit of an Infinite Series in Single-Variable Calculus
An infinite series is defined as the limit of its sequence of partial sums: the series converges to a value L if and only if this limit exists, and diverges otherwise. This principle belongs to the t…