Infinite series partial sums

This potential ( )(r) is infinite if the central cell is not neutral, i.e., the sum of qi is not zero, and otherwise is an example of a conditionally convergent infinite series, as discussed above, so a careful treatment is necessary. The potential depends on the order of summation, that is, the order in which partial sums over n are computed. For example, for positive integers K, define ( )s (r) as... 

Partial Sums of Infinite Series, and How They Grow Calculus textbooks devote much space to tests for convergence and divergence of series that are of little practical value, since a convergent... 

From a series, one can form an infinite sequence by defining the partial sum as follows ... 

The two questions that we generally ask about an infinite constant series are (1) Does the series converge (2) What is the value of the series if it does converge Sometimes it is difficult to find the value of a convergent infinite series, and we then might ask how well we can approximate the series with a partial sum. 

In mathematics, a sequence is a set of quantities ordered by integers. A series is the sum of a sequence. A series can have an infinite number of terms (n oo), or it can be a partial sum of n terms. In a geometric series, each term is the product of the preceding term times x ... 

There is no general method for finding the value of every convergent infinite series, but some series can be summed by finding an appropriate method. If you cannot find a mefhod to sum a particular infinite series, you might be able to approximate a convergent series with a partial sum. 

It is easy to determine the difference between the entire sum and a partial sum using this equation. With some other convergent infinite series, we might try to apply Eq. (10.5) as a rough measure of the error in approximating shy Sn-... 

In the preceding paragraphs of this section we have summed the terms arising from the partial expansion of the exponentials occurring in the coefficients of the powers of particle concentrations to obtain a series of multiple infinite sums, the terms of which are convergent. The terms in S(R) are of the same form as those in the Mayer solution theory, apart from replacement of integration by summation and the fact that mu differs from the solution value because of the discreteness of the lattice. The evaluation of wi - is outlined in the next section. It is found that the asymptotic form is... 

---