If ∑an is convergent and (bn)n ≥0 is a sequence of real numbers, which is monotone and bounded,

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If ∑an is convergent and (bn)n ≥0 is a sequence of real...

If $\sum a_{n}$ is convergent and $\left(b_{n}\right)_{n \geq 0}$ is a sequence of real numbers, which is monotone and bounded, then the series $\sum a_{n} b_{n}$ is convergent (P.G.L. DIRICHLET, 1863).

Key Concepts

Convergence of Infinite Series

This concept refers to the idea that an infinite series converges if the sequence of its partial sums approaches a finite limit as the number of terms increases indefinitely. Understanding convergence is central in analysis, as it determines whether the sum of an infinite number of terms yields a meaningful, finite result.

Monotonic Sequences

A monotonic sequence is one that is either entirely non-increasing or non-decreasing. Monotonicity offers simplicity in analysis because such sequences exhibit a predictable behavior, making it easier to assess their limits or boundedness.

Bounded Sequences

A sequence is bounded if there exists a real number that serves as an upper limit and another that serves as a lower limit such that all terms of the sequence lie between these two values. Boundedness is a crucial property in convergence tests since it guarantees that the sequence does not diverge to infinity.

Dirichlet's Test

Dirichlet's Test is a criterion for determining the convergence of series formed by the term-by-term product of two sequences. It stipulates that if one sequence has bounded partial sums and the other sequence is monotonically decreasing (or more generally, monotonic and bounded), then their product series converges. This test is particularly useful when dealing with series that are not absolutely convergent.

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