sumk1infty

Sum from k = 1 to infinity, written as ∑_{k=1}^∞ a_k, denotes the infinite series formed by adding the consecutive terms a_k starting at k = 1. The value of the series is defined as the limit of its partial sums S_n = ∑_{k=1}^n a_k as n grows without bound. If this limit exists, the series is convergent and its sum is that limit; if the limit does not exist, the series diverges.

Convergence can be studied with several criteria and tests. If ∑ |a_k| converges, the series is absolutely convergent and its sum is independent of the order of terms. If ∑ a_k converges but ∑ |a_k| diverges, the series is conditionally convergent. Common tests include the ratio test, root test, comparison test, integral test, and the alternating series test, which applies to series with alternating signs and decreasing terms tending to zero.

Several standard examples illustrate these ideas. A geometric series ∑ a r^{k-1} converges if |r| < 1 and then sums to a/(1−r). The harmonic series ∑ 1/k diverges, while the p-series ∑ 1/k^p converges if p > 1 and diverges if p ≤ 1. The alternating harmonic series ∑_{k=1}^∞ (-1)^{k+1}/k converges to ln 2. The series ∑_{k=1}^∞ 1/k^2 converges to π^2/6, illustrating that some sums have closed-form values albeit not always in elementary terms.

In practice, the infinite sum is a foundational concept in analysis, used to represent functions, define special constants, and express solutions in physics and engineering. Not all series have simple closed forms, but their convergence properties are essential for rigorous mathematical work.