"In mathematics, a series is the sum of the terms of an infinite sequence of numbers."
A series that has a finite sum.
Series: A series is a sum of an infinite sequence of numbers. Understanding the concepts of series is fundamental to the study of convergent series.
Sequences: A sequence is a list of numbers in a defined order. Understanding the concepts of sequences is fundamental to the study of convergent series.
Terms and Summands: The terms and summands of a series refer to the individual numbers which make up the sequence or sum.
Convergence and Divergence: Convergence and divergence refer to whether a series can be summed to a finite or infinite value, respectively. Understanding these concepts is essential when studying series and sequences.
Absolute Convergence: A series is said to be absolutely convergent if the series formed by taking the absolute value of each term converges.
Ratio Test: The ratio test is a method of determining the convergence or divergence of a series by comparing the ratio of the terms in the series.
Root Test: The root test is a method of determining the convergence or divergence of a series by comparing the nth root of the absolute value of the terms in the series.
Comparison Test: The comparison test is a method of comparing the convergence or divergence of two series by looking at the relative size of their terms.
Alternating Series: An alternating series is a series whose terms have alternating signs. Understanding the behavior of alternating series is essential when studying convergent series.
Power Series: A power series is a series where the terms are powers of a variable. Understanding this type of series is important for many applications in mathematics and physics.
Taylor Series: The Taylor series is a power series that represents a function as an infinite sum of terms. Understanding this type of series is important for many applications in calculus.
Fourier Series: The Fourier series is a way of representing periodic functions as an infinite sum of sine and cosine functions. Understanding this type of series is important for many applications in signal processing and physics.
Taylor’s Theorem: Taylor’s theorem gives a formal way of expressing a function as an infinite sum of terms based on its derivatives. Understanding this theorem is important for many applications in calculus and physics.
Cauchy’s Theorem: Cauchy’s theorem gives conditions under which a series converges. Understanding this theorem is important for many applications in mathematics and physics.
Geometric Series: A series in which each term is multiplied by a fixed factor known as the common ratio.
Telescoping Series: A series in which all terms except the first and last cancel out.
p-series: A series of the form Σ(1/n^p), where p is a positive constant.
Alternating Series: A series in which the signs of the terms alternate.
Absolute Convergence Series: A series in which the sum of the absolute values of all the terms is finite.
Ratio Test Series: A series in which the limit of the ratio of consecutive terms approaches a finite limit.
Integral Test Series: A series in which the terms can be expressed as the integral of a continuous function.
Power Series: A series in which the terms have an increasing power of some variable.
Taylor and Maclaurin series: A series in which a function is represented as a sum of infinite terms of its derivatives.
"A series S that is denoted S = a0 + a1 + a2 + ⋯ = ∑k=0∞ ak."
"The nth partial sum Sn is the sum of the first n terms of the sequence."
"Sn = ∑k=1n ak."
"A series is convergent (or converges) if the sequence of its partial sums tends to a limit."
"When adding one ak after the other in the order given by the indices, one gets partial sums that become closer and closer to a given number."
"A series converges if there exists a number ℓ such that for every arbitrarily small positive number ε, there is a (sufficiently large) integer N such that for all n ≥ N, |Sn - ℓ| < ε."
"If the series is convergent, the (necessarily unique) number ℓ is called the sum of the series."
"The same notation ∑k=1∞ ak is used for the series, and, if it is convergent, to its sum."
"This convention is similar to that which is used for addition: a + b denotes the operation of adding a and b as well as the result of this addition, which is called the sum of a and b."
"Any series that is not convergent is said to be divergent or to diverge."
"A series that does not converge."
"The nth partial sum Sn is the sum of the first n terms of the sequence."
"A series is convergent (or converges) if the sequence of its partial sums tends to a limit."
"When adding one ak after the other in the order given by the indices."
"If the series is convergent, the (necessarily unique) number ℓ is called the sum of the series."
"The sum of a series is denoted ∑k=1∞ ak."
"The same notation ∑k=1∞ ak is used for the series, and, if it is convergent, to its sum."
"A series converges if there exists a number ℓ such that for every arbitrarily small positive number ε, there is a (sufficiently large) integer N such that for all n ≥ N, |Sn - ℓ| < ε."
"One gets partial sums that become closer and closer to a given number."