"In mathematics, a divergent series is an infinite series that is not convergent, meaning that the infinite sequence of the partial sums of the series does not have a finite limit."
A series that does not have a finite sum.
Arithmetic sequence: A sequence that follows a constant difference between consecutive terms.
Geometric sequence: A sequence that follows a constant ratio between consecutive terms.
Summation notation: A notation used to represent the sum of a sequence.
Partial sum: The sum of the first n terms of a sequence.
Convergent sequence: A sequence that has a finite limit as n approaches infinity.
Divergent sequence: A sequence that does not have a finite limit as n approaches infinity.
Limit of a sequence: The value that a sequence approaches as n approaches infinity.
Cauchy criterion: A criterion that determines if a sequence converges by checking if the terms get arbitrarily close to each other.
Integral test: A test that checks if a series converges by comparing it to an integral.
Ratio test: A test that checks if a series converges by computing the limit of the ratio of consecutive terms.
Comparison test: A test that checks if a series converges by comparing it to another series.
Alternating series: A series with alternating signs.
Absolute convergence: A series that converges regardless of the order of its terms.
Conditional convergence: A series that only converges under certain orderings of its terms.
Power series: An infinite series in which each term is a power of the variable.
Taylor series: A power series that approximates a function.
Fourier series: A series that represents a periodic function in terms of sines and cosines.
Alternating Series: A series where the signs of the terms alternate in a predictable way such as (-1)^n or (-1)^(n+1).
Harmonic Series: A series of the form 1/1 + 1/2 + 1/3 + 1/4 + ... which is known to be divergent.
Geometric Series: A series of the form a + ar + ar^2 + ar^3 + ... where r is a common ratio and a is the first term.
P-Series: A series of the form 1/n^p where p is any positive number. This series converges for p > 1 and diverges for p <= 1.
Factorial Series: A series of the form 1/1! + 1/2! + 1/3! + ... where n! means n factorial.
Exponential Series: A series of the form e^x = 1 + x + x^2/2! + x^3/3! + ... where e is Euler's number, and x is any real number.
Power Series: A series of the form ∑a_n(x-a)^n where a_n is the nth coefficient and a is a constant.
"If a series converges, the individual terms of the series must approach zero."
"Convergence is a stronger condition: not all series whose terms approach zero converge."
"A counterexample is the harmonic series: 1 + 1/2 + 1/3 + 1/4 + ..."
"The divergence of the harmonic series was proven by the medieval mathematician Nicole Oresme."
"In specialized mathematical contexts, values can be objectively assigned to certain series whose sequences of partial sums diverge, in order to make meaning of the divergence of the series."
"A summability method or summation method is a partial function from the set of series to values."
"For example, Cesàro summation assigns Grandi's divergent series: 1 - 1 + 1 - 1 + ... the value 1/2."
"Cesàro summation is an averaging method, in that it relies on the arithmetic mean of the sequence of partial sums."
"Other methods involve analytic continuations of related series."
"In physics, there are a wide variety of summability methods."
"These are discussed in greater detail in the article on regularization."
"A convergent series is an infinite series whose terms approach zero and the infinite sequence of the partial sums has a finite limit."
"No, if a series is divergent, the individual terms of the series do not approach zero."
"No, the harmonic series is an example of a divergent series."
"The divergence of the harmonic series was proven by Nicole Oresme."
"In specialized mathematical contexts, values can be objectively assigned to certain series whose sequences of partial sums diverge, in order to make meaning of the divergence of the series."
"Cesàro summation is based on the arithmetic mean of the sequence of partial sums."
"Yes, analytic continuations of related series can be used as methods."
"These methods are discussed in greater detail in the article on regularization."