Sequences, series, summability Definition
Geometric sequence
Builds on Real sequence
Definition
Let and be real numbers. The geometric sequence with initial term and common ratio is the real sequence . Its terms are , and each term is times the term before it.
Remarks
Many texts require and , so that the quotient of two consecutive terms is always . The sequence has and . The sequence has . For , the sequence converges to . For and , it tends to infinity. The sum of the terms is the geometric series.
Reading path
To understand this page from the ground up, read these 3 pages in order. The foundations come first.
Show the path as a graph
- no evidence level
- Sequence definition
- Real sequence definition
- Geometric sequence definition
Where this leads
114 pages build on this one.
- Power series are differentiable inside the radius of convergence
- Natural logarithm
- Euler–Mascheroni constant
- The logarithm grows slower than any power
- Derivative of sine
- 1 + x is at most exp x
- cos pi = -1
- sin x / x tends to 1
- Geometric series
- Inverse trigonometric functions
- Power series of sine
- Leibniz formula for pi
- Integral of exp(-x) over the half line
- Pythagorean trigonometric identity
- log x is at most x - 1
- Inverse hyperbolic functions
- Integral of cosine
- Sum of a finite geometric series
- Root test
- Derivative of arctangent
- Harmonic numbers grow like log n
- Integral of the exponential function
- Power series of the exponential function
- cosh^2 - sinh^2 = 1
- Cauchy–Hadamard theorem
- Derivatives of the hyperbolic functions
- Derivative of the logarithm
- Divergence part of the ratio test
- Power series of cosine
- Arcsine
- pi
- Exponential growth beats polynomial growth
- The exponential function is convex
- Gaussian integral
- Basel problem
- Exponential of a sum
- Trigonometric substitution
- Hyperbolic cosine
- The exponential function is its own derivative
- The gamma function extends the factorial
- e as the limit of (1 + 1/n)^n
- Sum of the reciprocals of the factorials
- Derivative of arcsine
- Addition formulas for hyperbolic functions
- Decimal expansion as a series
- Hyperbolic sine
- Integral of sine
- Hyperbolic functions
- Logarithm of a quotient
- Ratio formula for the radius of convergence
- Integral of 1/x
- Logarithm of a product
- Stirling's formula
- The exponential function is strictly increasing
- Radius of convergence of the binomial series
- Derivative of x^a
- e
- Exponentials dominate powers in sequences
- Hyperbolic tangent
- Sine is 2π-periodic
- The exponential function is positive
- The exponential series has infinite radius of convergence
- The sequence r^n tends to zero for |r| < 1
- Irrationality of pi
- Ratio test
- pi is greater than 3
- Arccosine
- Logarithm is the inverse of the exponential
- Exponential function
- Power-reduction formula for cosine
- Derivative of tangent
- Power series of arctangent
- Real exponentiation
- The sequence r^n tends to infinity for r > 1
- Derivative of cosine
- Logarithm of a power
- Improper integral of x^p on an unbounded interval
- Irrationality of e
- Integration by partial fractions
- x^n converges pointwise but not uniformly on [0, 1]
- Mercator series
- The logarithm is strictly concave
- Weierstrass factorization theorem
- Convergence of the Neumann series for Volterra kernels
- Mehta integral
- Weierstrass product for the gamma function
- Bertrand's test
- The exponential map from the additive reals to the positive reals
- Mertens' theorems
- Ramanujan's series for 1/pi
- Neumann series
- Weierstrass product
- Kummer's test
- Frullani integral
- Gauss's test
- Log-convexity of the gamma function
- Raabe's test
- Logarithmically convex function
- Reduction of a first-kind Volterra equation to the second kind
- Liouville–Neumann series
- Laplace transforms of sine and cosine
- Spectral radius formula
- Hadamard factorization theorem
- Viète's formula
- Bohr–Mollerup theorem
- Iterated exponentiation
- An infinite product converges iff its series of logarithms converges
- Trigonometric integrals
- Basel problem via Fourier series
- Unique solvability of Volterra equations of the second kind
- Laplace transform of the exponential function
- Function of exponential order
- Barnes G-function
- Chudnovsky algorithm
Further reading
- Web https://en.wikipedia.org/wiki/Geometric_progression, lead section
About this page
| Tag | 0MNM54WB99 |
|---|---|
| URI | m2://0MNM54WB99 |
| Aliases | geometric-progression |
| Kind | definition |
| Writing | W1 stated |
| MSC | 40A05 |
| Defines | Geometric sequence |
| Uses | Real sequence |
Markdown · JSON · Edit on GitHub
What this page needs for W2
- edge instance_of in: one example (0 of 1)
- edge not_instance_of in: one non-example that fails one clause (0 of 1)
The writing levels say how complete the text of a page is. They do not measure its truth.