Ads
related to: sum of series examples geometry problems and answerskutasoftware.com has been visited by 10K+ users in the past month
Search results
Results from the WOW.Com Content Network
The geometric series on the real line. In mathematics, the infinite series 1 / 2 + 1 / 4 + 1 / 8 + 1 / 16 + ··· is an elementary example of a geometric series that converges absolutely. The sum of the series is 1. In summation notation, this may be expressed as
An infinite series of any rational function of can be reduced to a finite series of polygamma functions, by use of partial fraction decomposition, [8] as explained here. This fact can also be applied to finite series of rational functions, allowing the result to be computed in constant time even when the series contains a large number of terms.
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 .
Examples of a geometric sequence are powers r k of a fixed non-zero number r, such as 2 k and 3 k. The general form of a geometric sequence is , , , , , … where r is the common ratio and a is the initial value. The sum of a geometric progression's terms is called a geometric series.
A famous example of an application of this test is the alternating harmonic series = + = + +, which is convergent per the alternating series test (and its sum is equal to ), though the series formed by taking the absolute value of each term is the ordinary harmonic series, which is divergent.
Those methods work on oscillating divergent series, but they cannot produce a finite answer for a series that diverges to +∞. [6] Most of the more elementary definitions of the sum of a divergent series are stable and linear, and any method that is both stable and linear cannot sum 1 + 2 + 3 + ⋯ to a finite value (see § Heuristics below).
The sum of the series is approximately equal to 1.644934. [3] The Basel problem asks for the exact sum of this series (in closed form ), as well as a proof that this sum is correct. Euler found the exact sum to be π 2 / 6 {\displaystyle \pi ^{2}/6} and announced this discovery in 1735.
This is a particular case of the sum of the reciprocals of any geometric series where the first term and the common ratio are positive integers. If the first term is a and the common ratio is r then the sum is r / a (r − 1) . The Kempner series is the sum of the reciprocals of all positive integers not containing the digit "9" in base 10.
Ads
related to: sum of series examples geometry problems and answerskutasoftware.com has been visited by 10K+ users in the past month