Continuous functions over real-number closed intervals can be absolutely and uniformly convergent series of polynomials {convergent polynomial series}.
Series {monotone series} can be single-valued, have bounds, be piecewise continuous, and have finite numbers of discontinuities, maxima, and minima.
Sequences {null sequence} can have zero as limit. Sequence-element absolute values can be less than positive rational numbers.
Sequences {ordered sequence} can have one-to-one correspondence with positive integers.
Sequences {sum of sequences} can have terms that are sums of terms of two other sequences. For sequences that converge, limit of sum sequence is sum of original-sequence limits.
Sequences {product sequence} can have terms that are products of terms of two other sequences. For sequences that converge, limit of product sequence is product of original sequence limits.
Sequences {quotient sequence} can have terms that are quotients of terms of two other sequences. For sequences that converge, limit of quotient sequence is quotient of original sequence limits.
Number series can have general term 2^(2^n) + 1, for n = 0, 1, 2, ... {Fermat's numbers} {Fermat numbers}.
Fibonacci sequences have consecutive-number ratios that approach golden section, (1 + 5^0.5) / 2 {Fibonacci ratio}.
pi = 4 * (1 - 1/3 + 1/5 - 1/7 + 1/9 - ...) {Gregory's series} {Gregory series}.
Arithmetic progression can have first term one and common difference n - 2, where n is number of polygon sides {polygonal number}.
n can be three, so sequence is 1, 3, 6, 10, 15, 21, ... {triangular number}.
n can be four, so sequence is 1, 4, 9, 16, 25, ... {square number, polygonal}.
n can be five, so sequence is 1, 5, 12, 22, ... {pentagonal number}.
For n, sequence is l, n, 3*n - 3, 6*n - 8, ... {n-gonal number}.
general
In general, 0.5 * (r + 1) * (r*n - 2*r + 2), where r is whole number, makes polygonal numbers.
Difference {common difference} between consecutive sequence terms can be constant {arithmetic progression}. General term is a(n) = a(1) + d * (n - 1), where d is common difference. First-n-terms partial sum is n times average of first and last terms: Sn = n * (a(1) + a(n)) / 2, where a(k) is general term, and n is number of terms.
Consecutive sequence-term ratios {common ratio} can be constant {geometric progression}. General term is a(n) = a(1) * r^(n - 1), where r is common ratio, and n is number of terms. Partial sum is Sn = a(1) * (1 - r^n) / (1 - r), where r is common ratio, and n is number of terms.
Difference between reciprocals of successive sequence terms can be constant {harmonic progression}. General term a(n) is 1 / a(n) = 1 / (a(1) + d * (n - 1)), where d is difference.
Sequence general term can be a(n) * x^n {power series}|, where a are rational coefficients, and n is number of terms. Power series converges if independent variable equals zero. x^n / n! converges at all x.
1 + a1 * u + a2 * u^2 / 2! + ... {associated series}, where ai are series coefficients.
Power series {expansion, function} {expansion, series} can replace function or integral.
e^x = 1 + x + x^2 / 2! + x^3 / 3! + ... {exponential series}.
sinh(x) = x + x^3 / 3! + x^5 / 5! + ... {hyperbolic expansion}. cosh(x) = 1 + x^2 / 2! + x^4 / 4! + ...
1 + a*b*x / c + (a * (a + 1) * b * (b + 1) * x^2) / (2 * c * (c + 1)) + ... {hypergeometric series} can converge at x. If x = 1, function equals (gamma(c) * gamma(c - a - b)) / (gamma(c - a) * gamma(c - b)).
1 + x + 2! * x^2 + 3! * x^3 + ... {Laguerre series}.
ln(1 + x) = x - x^2 / 2 + x^3 / 3 - x^4 / 4 + ... {logarithmic series}, where -1 <= x <+ +1. ln(N + 1) - ln(N) = 2 * (1 / (2*N - 1) + 1 / (3 * (2*N - 1)^3) + 1 / (5 * (2*N - 1)^5) + ...).
f(b) = f(0) + (sum from i = 0 to i = n - 1 of (function ith derivative at 0) / i!) * b^i + error term {Maclaurin expansion}. Maclaurin expansion is Taylor series with a = 0.
Fractions {radix fraction} can be sums of positive common fractions a/r + b / r^2 + c / r^3 + ..., where r is a number, and a b c are integers. Common fractions can be radix fractions, using any radix in any number system.
f(b) = f(a) + (sum from i = 0 to i = n - 1 of (function ith derivative at a) / i!) * (b - a)^i + error term {Taylor's theorem} {Taylor theorem} {Taylor series}. For two variables, Taylor series has twice as many terms {Taylor expansion}.
Over intervals, periodic functions can be infinite sine and cosine series {trigonometric series}.
Trigonometric series {Fourier series} {Fourier integral} can represent function over interval: (1 / pi) * (integral from -a = infinity to a = +infinity of F(a) * da) * (integral from a = 0 to a = infinity of cos(q * (x - a)) * da). Complex waveforms over time or position can be finite or infinite series of harmonic sine and cosine waves {Fourier analysis}|: f(x) = (2 * pi)^-0.5 * (integral from -infinity to +infinity of g(p) * e^(i*x*p) * dp), where g(p) is density {Fourier transform, series}. Complex function can have g(p) = 0 for p >= 0 {positive frequency function}.
convergence
If Fourier series is single-valued, has a bound, is piecewise continuous, and has finite numbers of discontinuities, maxima, and minima {Dirichlet condition}, it converges.
domain
Domain can be circle whose circumference is period or wavelength.
range
Series can be on unit circle in complex plane.
theorem
Fourier-series coefficients can exist and have properties {Parseval's theorem}.
Sines and cosines of Fourier series can be complex exponentials {Laurent series}: F(z) = F(e^i * a * x) = sum from -infinity to + infinity of Ar * z^r, where z is complex number, Ar is general term, and r is convergence radius. Laurent series has convergence annulus on Riemann sphere. Convergence circle can be for positive-term sum {positive frequency part}. Convergence circle can be for negative-term sum {negative frequency part}.
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Date Modified: 2022.0225