Terms can add or multiply in sequence {series, mathematics} {mathematical series}. Series has sum or product. Series has general term and follows rule. For example, 1 + 2 + 3 + ... is series that has general term n and follows the rule that next term is one higher than previous term. Series sum through nth position is n * (n + 1) / 2.
For series, a formula {recursion formula} gives next term from previous term.
For series, sum {partial sum} adds first term through nth term: Sn = sum from k = 1 to k = n of general term a(k). Other sums {averaging partial sums} {Holder summability} {Cesaro sum} include converting divergent series to continued fractions, or vice versa.
For series, a product {partial product} multiplies first term through nth term: Pn = product from k = 1 to k = n of general term a(k).
Power series can have error term {error term} {remainder, series}, which can have Taylor error term form and other forms {Schlömilch form} {Lagrange form} {Cauchy form}.
Series {sequence, mathematics} {mathematical sequence} can have numbers or terms {ordered term} in sequence. Sequence has general term {general term, sequence} and follows rule. Example sequence is x, 2*x, 3*x, ... General term a(n) = n * x, where n is term position. The rule is that n increases by one.
separator
Commas separate sequence terms, as in 1, 2, 3, ...
types
Sequences can descend, ascend, or alternate. Sequence terms can approach number {convergent sequence}. Sequence terms can approach infinity {divergent sequence}. Sequence can be neither convergent nor divergent {indefinite sequence}. Sequence can increase, decrease, increase, and so on {oscillating sequence} {alternating sequence}.
Two sequences are equal {coincident sequence} if and only if all corresponding sequence terms are equal.
The only sequence whose first term equals zero, and whose n+1th term equals zero if nth term equals zero, is sequence of zeroes {induction axiom}. The only sequence whose first term equals one, and whose n+1th term equals k + 1 if nth term equals k, is the positive-integer sequence. The positive-integer series is the only sequence that has the number one and contains the positive integers.
If successive-term absolute value is less than previous-term absolute value, series converges {absolutely convergent} {convergence, series}.
Series can be convergent even if successive-term absolute value is not less than previous-term absolute value {conditionally convergent}. Conditionally convergent series can rearrange to make sum be any number.
constant times sequence
If sequence converges, limit of constant times sequence is constant times sequence limit.
uniform
Absolute value of partial sum S(n) minus sum from x = 1 to x = n of S(n) * x can be less than small value, for all x {uniform convergence}.
Convergent-power-series partial-sum limit is partial sum, if convergence radius replaces general-term independent variable {Abel summability}.
Divergent series {asymptotic series} can represent functions and evaluate integrals. In such divergent series, nth-term error is less than n+1th-term absolute value, so error becomes less as term number increases.
Absolute value of each sequence term can be less than or equal to a constant {bounded sequence}.
If ratio between next-term absolute value and previous-term absolute value is greater than or equal to one, sequence diverges {divergence, series}.
semiconvergence
Divergent series {semiconvergent series} can have nth-term error less than n+1th-term absolute value, so error decreases faster than terms increase. Though they diverge, semiconvergent series can evaluate integrals, because sum is finite. Useful asymptotic series is f(x) = a0 + a1/x + a2/(x^2) + ... If x is large, limit of x^n * (f(x) - series) = 0. x approaches 0 if 1/(x^n) * (f(x) - series) = a(n). Other ideas about asymptotic series include {Birkhoff's theorem} {Birkhoff's connection formula} {WKBJ solution} {Airy's integral}.
If sequence successive term is less than previous term, and if general term is always less than constant, sequence has limit {limit, sequence} {sequence limit} {limit, series} less than or equal to constant.
Summable series can make convergent series {Tauberian theorem}.
Independent variable can have value {radius of convergence} {convergence radius} greater than zero at which power series changes from convergence to divergence. Power series, power-series differential, and power-series integral have same convergence radius.
For complex-number power series, if complex number lies within a complex-plane circle {convergence circle} {circle of convergence} centered on zero, with no singularities, series converges. If complex number lies outside a complex-plane circle, series diverges.
Laurent series has complex number that lies within annulus in complex plane {convergence annulus} {annulus of convergence}.
Independent variable can have values {region of convergence} {convergence region, series} for which series converges.
If and only if absolute value of difference between successive partial sums is less than small value {Cauchy convergence criterion}, series converges.
Sequence general term can be less than or equal to constant times second-sequence general term {comparison test}. If second sequence diverges, first sequence diverges. If second sequence converges, first sequence converges.
If second sequence converges, and if second-sequence general term divided by sequence general term has limit, first sequence converges.
If second sequence diverges, and second-sequence general term divided by sequence general term has limit or if quotient is infinite, first sequence diverges.
If limit of quotient of sequence general terms does not equal zero, both sequences either diverge or converge.
Integral from x = 0 to x = a of (f(x) * sin(u*x) / sin(x)) * dx, and integral from x = a to x = b of (f(x) * sin(u*x) / sin(x)) * dx {Dirichlet integral}, where b > a > 0, can show convergence.
Sequences converge if and only if integral of general term, from x equals some value to x equals infinity, exists {integral test}.
Alternating sequences can converge {Leibniz's test} {Leibniz test}.
If successive-term to previous-term ratio limit is less than one, sequence converges {ratio test}. If successive-term to previous-term ratio limit is greater than one, sequence diverges. If successive-term to previous-term ratio limit is one, sequence can converge or diverge. If general-term limit equals zero, successive-term to previous-term-ratio absolute-value limit is less than one. Generalized ratio test {d'Alembert's test} exists.
To average sequence terms {arithmetic mean}, add all terms and divide by number of terms: (sum from k = 1 to k = n of a(k)) / n, where a(k) is general term, k is sequence position, and n is number of terms.
To average sequence terms {geometric mean}|, multiply all sequence numbers to get product, and then take product term-number root: (product from k = 1 to k = n of a(k))^(1/n), where a(k) is general term, k is sequence position, and n is term number.
To average sequence terms {harmonic mean}|, divide number of terms by sum of sequence-term reciprocals: n / (sum from k = 1 to k = n of 1/a(k)), where a(k) is general term, k is sequence position, and n is number of terms.
Arithmetic mean {mean, series} can be greater than or equal to geometric mean, which is greater than or equal to harmonic mean {Cauchy's principle} {Cauchy principle}.
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