Calculate a Series
Calculator for an infinite series, which converges to a fixed value. The result will be reached with a given accuracy. The higher the accuracy, the longer takes the calculation. A series is a sum with the start value 0 and theoretically an infinite amount of steps. Here, a value of the series is regarded as a result, if five values in a row with the given accuracy are equal. If the upper limit is reached, then the last value will be shown as an intermediary result.
As running variable, which is increased by 1 in each step, i is used. Only this variable may occur in the sum term. The basic arithmetic operations + - * / are allowed, as well as the power function, like pow(1/2#i) for (1/2)i. Further allowed functions are sin(), cos(), tan(), asin(), acos(), atan(), log() for the natural logarithm and fac() for the factorial. Also, the constants e and pi can be used.
Example: a series Σ qi is called geometric series, if q is a constant between 0 and 1. For the given series (1/2)i = 1+1/2+1/4+1/8+1/16+..., the result is 2. With three decimal places, it takes 14 steps to determine it, with 15 decimal places, 54 steps are needed.
Series often tend to infinity, then no result value is reached. To have a series converging, heading to a fixed value, the single summands (the parts that are added to each other) can e.g. decline exponentially, like at the geometric series.
Series appeared before the 17th century, but only slowly became established from this time onwards. They made a decisive contribution to the development of infinitesimal calculus by Gottfried Wilhelm Leibniz and Isaac Newton. The first of the two discovered a series expansion for the number π.
Taylor series and Taylor expansion play a special and important role in mathematics.