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for the infinite series. Note that if the function () is increasing, then the function () is decreasing and the above theorem applies.. Many textbooks require the function to be positive, [1] [2] [3] but this condition is not really necessary, since when is negative and decreasing both = and () diverge.
In mathematics, a series is the sum of the terms of an infinite sequence of numbers. More precisely, an infinite sequence (,,, …) defines a series S that is denoted = + + + = =. The n th partial sum S n is the sum of the first n terms of the sequence; that is,
The first block is a unit block and the dashed line represents the infinite sum of the sequence, a number that it will forever approach but never touch: 2, 3/2, and 4/3 respectively. A geometric progression , also known as a geometric sequence , is a mathematical sequence of non-zero numbers where each term after the first is found by ...
A sequence of functions () converges uniformly to when for arbitrary small there is an index such that the graph of is in the -tube around f whenever . The limit of a sequence of continuous functions does not have to be continuous: the sequence of functions () = (marked in green and blue) converges pointwise over the entire domain, but the limit function is discontinuous (marked in red).
In the mathematical field of real analysis, the monotone convergence theorem is any of a number of related theorems proving the good convergence behaviour of monotonic sequences, i.e. sequences that are non-increasing, or non-decreasing.
When successive powers of a matrix T become small (that is, when all of the entries of T approach zero, upon raising T to successive powers), the matrix T converges to the zero matrix.
The 2-8-8-2 was a design largely limited to American locomotive builders. The last 2-8-8-2 was retired in 1962 from the N&W's roster, two years past the ending of steam though steam was still used on steel mill lines and other railroads until 1983. Other equivalent classifications are: UIC classification: (1′D)D1′ French classification: 140+041
In mathematics, the Laurent series of a complex function is a representation of that function as a power series which includes terms of negative degree. It may be used to express complex functions in cases where a Taylor series expansion cannot be applied.