![]() The sum( function returns the sum of the terms of any list. If it is convergent, the partial sums can also help estimate the sum of the series. Partial sums can be computed with the sum function and may be used to help explore whether or not an infinite series converges. The TI-83's sum( function and the Sequence Graphing mode are useful tools in understanding the sequence of partial sums of series. Otherwise, the infinite series does not have a sum and it is divergent. ![]() If the sequence of partial sums for an infinite series converges to a limit L, then the sum of the series is said to be L and the series is convergent. For the series given above, the sequence of partial sums is ![]() The partial sums of a series form a new sequence, which is denoted as. The first four partial sums of the associated infinite series are computed below, where s k represents the sum of the first k terms of the sequence.Įach of the results shown above is a partial sum of the seriesĭefining the Sequence of Partial Sums of a Series Suppose an infinite sequence is defined by When working with infinite series, it is often helpful to examine the behavior of the partial sums. However, when the series has an infinite number of terms the summation is more complicated and the series may or may not have a finite sum.Ī partial sum of an infinite series is the sum of a finite number of consecutive terms beginning with the first term. If the series has a finite number of terms, it is a simple matter to find the sum of the series by adding the terms. These applications arise in many disciplines, especially physics and chemistry.Ī series, which is not a list of terms like a sequence, is the sum of the terms in a sequence. Series are used in many applications including integration, approximation, and the solution of differential equations. This lesson explores series and partial sums of infinite series. Lesson 23.2: Series and Sequences of Partial Sums Module 23 - Sequences and Series - Lesson 2 Module 23 - Sequences and Series
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