Kronecker's lemma

Kronecker's lemma converts convergence of a series into convergence to zero of weighted averages of its terms when the positive weights increase without bound.

Summation by parts: A discrete identity that transfers differences between two sequences, analogous to integration by parts. It rewrites weighted partial sums using the convergent series' partial sums.
Infinite series: The limit of the partial sums of an ordered sequence of terms. Kronecker's lemma begins with convergence of such a series.
Strong law of large numbers: A theorem stating that sample averages converge almost surely to the expected value under suitable assumptions. Kronecker's lemma converts convergence of normalized centered sums into convergence of averages.
Kronecker's theorem: A theorem in Diophantine approximation concerning density of sequences on tori. Its shared name can obscure that it is unrelated to this summability lemma.
Abel transformation: A finite-sum identity expressing a sum of products through partial sums and successive differences. This is the standard algebraic form used to prove Kronecker's lemma.
Partial sum: The sum of the first finitely many terms of a sequence or series. The lemma applies its increasing weights to partial sums of the given series.
Kolmogorov's convergence criterion: A theorem giving conditions for almost-sure convergence of series of independent random variables. Its series-convergence conclusion can feed directly into Kronecker's lemma.
Hardy's Tauberian theorem: A theorem connecting Abel summability with ordinary convergence under additional Tauberian conditions. Both results connect generalized summation behavior with ordinary limits.
Cauchy criterion: A criterion stating that a sequence converges exactly when its terms become arbitrarily close in sufficiently late tails. Tail control for a convergent series bounds the transformed weighted sums.
Sequence convergence: The property that sequence terms approach a finite limit as their indices increase. The lemma's conclusion is a limit of a sequence of normalized sums.

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