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Comparison test

Comparison test can mean:Limit comparison test, a method of testing for the convergence of an infinite series.Direct comparison test, a way of deducing the convergence or divergence of an infinite series or an improper…

Comparison test

Comparison test can mean:

Limit comparison test, a method of testing for the convergence of an infinite series.
Direct comparison test, a way of deducing the convergence or divergence of an infinite series or an improper integral.

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What is the ε, δ definition actually saying?

That you can make the output as close to the limit as anyone demands (within ε) by keeping the input close enough (within δ). It replaces "approaches" with a challenge-and-response that can be checked.

Why does the harmonic series diverge when its terms go to zero?

Because the terms shrink too slowly: group them as 1/3 + 1/4 > 1/2, 1/5 + … + 1/8 > 1/2, and so on: infinitely many halves.

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Diẹ̀ nínú Real Analysis

Sequences and their limits Convergence of series Improper integrals Taylor approximation and error Functions of a complex variable Continuity and differentiability, rigorously Sequences of functions and uniform convergence