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The following list includes a decimal expansion and set containing each number, ordered by year of discovery. The column headings may be clicked to sort the table alphabetically, by decimal value, or by set. Explanations of the symbols in the right hand column can be found by clicking on them.
The circumference of a circle with diameter 1 is π.. A mathematical constant is a number whose value is fixed by an unambiguous definition, often referred to by a special symbol (e.g., an alphabet letter), or by mathematicians' names to facilitate using it across multiple mathematical problems. [1]
When every term of a series is a non-negative real number, for instance when the terms are the absolute values of another series of real numbers or complex numbers, the sequence of partial sums is non-decreasing. Therefore a series with non-negative terms converges if and only if the sequence of partial sums is bounded, and so finding a bound ...
The geometric series is an infinite series derived from a special type of sequence called a geometric progression.This means that it is the sum of infinitely many terms of geometric progression: starting from the initial term , and the next one being the initial term multiplied by a constant number known as the common ratio .
This sum can be found quickly by taking the number n of terms being added (here 5), multiplying by the sum of the first and last number in the progression (here 2 + 14 = 16), and dividing by 2: (+) In the case above, this gives the equation:
The sum of the series is a random variable whose probability density function is close to for values between and , and decreases to near-zero for values greater than or less than . Intermediate between these ranges, at the values ± 2 {\displaystyle \pm 2} , the probability density is 1 8 − ε {\displaystyle {\tfrac {1}{8}}-\varepsilon } for ...
The series converges extremely slowly. Baillie [4] remarks that after summing 10 24 terms the remainder is still larger than 1. [9]The upper bound of 80 is very crude. In 1916, Irwin [10] showed that the value of the Kempner series is between 22.4 and 23.3, since refined to the value above, 22.92067...
Further information related to the numerical evaluation of Stieltjes constants may be found in works of Keiper, [23] Kreminski, [24] Plouffe, [25] Johansson [26] [27] and Blagouchine. [27] First, Johansson provided values of the Stieltjes constants up to n = 100000, accurate to over 10000 digits each (the numerical values can be retrieved from ...