The p-series for a real exponent p converges exactly when p is greater than one

Depends on

• The integral test: for $f \ge 0$ nonincreasing on $[0,\infty)$, $\sum_k f(k)$ converges if and only if the sequence $\bigl(\int_0^N f\bigr)_N$ is bounded, with $\int_0^N f \le \sum_{k<N} f(k) \le f(0) + \int_0^N f$
• Continuity and derivatives of positive-base real powers
• The exponent, product, quotient, and iterated-power laws for positive real bases and real exponents
• Order, continuity, range, and the product, quotient, and reciprocal laws for the natural logarithm
• The natural logarithm has derivative 1/x and equals the integral from 1 to x of 1/t
• The exponential tends to $+\infty$ at $+\infty$ and to $0$ at $-\infty$
• If a series converges then its terms tend to $0$

Used by

• Equal radii do not determine convergence on the boundary circle Counterexample
• Summable untruncated variances are not necessary Counterexample
• The log(1+x) power series diverges at x=-1 Counterexample
• Diagonal Dirichlet sine-mode heat model with constant forcing Definition
• Dirac comb Definition
• A bounded-noise Robbins–Monro scalar root Example
• A compact diagonal map has a closure-only feature Example
• A compact diagonal map with dense nonclosed range Example
• A Dirac mass has precisely sufficiently negative Sobolev order Example
• A power series with reciprocal-square coefficients converges uniformly on the closed unit disc Example
• Almost-sure conditional convergence Example
• Boundary-incompatible forcing reduces Strang order to five quarters in a Dirichlet modal model Example
• k⁻ᵃ membership in ℓᵖ Example
• Nonidentical strong law under summable normalized variances Example
• Rademacher-series threshold Example
• The harmonic complex power series diverges at 1 and converges conditionally at every other point of the unit circle Example
• The three series impose separate conditions Example
• Constant Dirichlet forcing is square summable but outside the heat-generator domain Lemma
• Derivative-window alias bounds including the P=2 knot branch Lemma
• Exact three-dimensional spline assignment aliases and FFT normalization Lemma
• Finite-measure growth increment lemma Lemma
• Separated Matérn sites have a positive Gram lower bound Lemma
• Smooth torus test functions have rapidly convergent Fourier pairings Lemma
• Stirling's factorial asymptotic holds up to a positive constant Lemma
• The Dirac comb of a full-rank lattice transforms to the dual comb Lemma
• Chebyshev–Lobatto interpolation has an algebraic weighted spectral-Sobolev rate Theorem
• Finite variance logarithmic rate for iid sums Theorem
• Mertens' first theorem for primes Theorem
• Poisson summation for a full-rank lattice Theorem
• Polynomial growth functions define tempered distributions Theorem
• Strong law under summable normalized variances Theorem

Dependency tree · two levels

45 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

• J. Lebl, Basic Analysis, Logarithm and Exponential (standard reference, not scraped)
• MIT OpenCourseWare 18.100B Real Analysis, Spring 2025 full lecture notes (standard reference, not scraped)