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Given an infinite series where the terms are positive numerical values, rather than algebraic expressions, (Q1) determine whether the series converges. If the series does converge, (Q2) how can I tell when the partial sum is within a fixed value of the limit, given only the terms so far. The point is to know when it is safe to stop computing terms.

For example, if terms of the series were $a_i={1\over 2^n}$, and the desired limit was 0.001, looking only at the first three terms, the partial sums are 0.5, 0.75, 0.875, ..., 1-1/1024=0.999023, and we are within the desired accuracy of 0.001.

In this example, since the form of the series, its convergence, and the limit are all known, it is trivial to answer the above questions. In practice, the terms are computed numerically by an external process, and we receive them one at a time. Also, it is not true that $a_{i+1}<a_i$ for all $i$, although it is usually true that ${\lim\atop{i\rightarrow\infty}}=0$ (a necessary condition for the series to converge).

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    $\begingroup$ If all we know about future terms is that they go to $0$, we do not even know whether the series converges, let alone error bounds. $\endgroup$ – André Nicolas Jul 28 '16 at 20:40
  • $\begingroup$ One can use extrapolation methods to obtain accurate estimates of the limit and about the convergence of the series. There methods are not fool proof, but they have been used with great success. Even when the task is to resum a wildly divergent asymptotic series given only the first few terms, very accurate results can be obtained. $\endgroup$ – Count Iblis Jul 28 '16 at 20:44
  • $\begingroup$ @André: Yes, convergence is Q1. Most of the time, the series will converge, though. $\endgroup$ – Oliver Jul 28 '16 at 22:54
  • $\begingroup$ @Count Iblis: Can you suggest a reference for some of these extrapolation methods? $\endgroup$ – Oliver Jul 28 '16 at 22:56
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When only a finite number of terms can be evaluated numerically, one has to resort to extrapolation methods to determine the properties of the function represented by the series. In general, it's helpful to replace a series with numerical coefficients $a_j$ by a function $S(z) = \sum_j a_j z^j$. The problem can then be recast as estimating $S(1)$,. Note that it's not necessary for the series to actually converge for $z=1$, what matters is that we've enough information from the series coefficients to approximately reconstruct $S(z)$.

A simple method that is widely used in practice is to use Padé approximants to approximately represent a series by a rational function. Applying the Padé method to the logarithmic derivative of the function often yields superior results (DLog-Padé method). A more sophisticated method to resum a series based on only a finite number of terms is to use differential approximants. Given a series $S(z)$ one finds polynomials $q_j(z)$ and $p(z)$ such that the differential equation:

$$\sum_{j=0}^n q_j(z)\left(z\frac{d}{dz}\right)^jS = p\tag{1}$$

is satisfied. The Padé method is then the special case for $n = 0$ while the DLog-Padé method is homogeneous $n=1$ case ($p(z)$ chosen to be the zero polynomial).

If the series $S(z)$ is given exactly as a function that satisfies a finite order differential equation with polynomial coefficients (a so-called "D-finite" function), then you can always reconstruct this function by solving for the coefficients of the polynomials in (1), which amounts to solving a linear system of equations.

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