Among the classical divergent series, is relatively difficult to manipulate into a finite value. Many
summation methods are used to assign numerical values to divergent series, some more powerful than others. For example,
Cesàro summation is a well-known method that sums
Grandi's series, the mildly divergent series , to .
Abel summation is a more powerful method that not only sums Grandi's series to , but also sums the trickier series to . Unlike the above series, is not Cesàro summable nor Abel summable. Those methods work on oscillating divergent series, but they cannot produce a finite answer for a series that diverges to +∞. 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 to a finite value . More advanced methods are required, such as
zeta function regularization or
Ramanujan summation. It is also possible to argue for the value of using some rough heuristics related to these methods.
Heuristics 's first notebook describing the "constant" of the series
Srinivasa Ramanujan presented two derivations of "" in chapter 8 of his first notebook. The simpler argument proceeds in two steps, as follows. The first key insight is that the series of positive numbers closely resembles the
alternating series The latter series is also divergent, but it is much easier to work with; there are several classical methods that assign it a value, which have been explored since the 18th century. In order to transform the series into , one can subtract 4 from the second term, 8 from the fourth term, 12 from the sixth term, and so on. The total amount to be subtracted is , which is 4 times the original series. These relationships can be expressed using algebra. Whatever the "sum" of the series might be, call it Then multiply this equation by 4 and subtract the second equation from the first: \begin{alignat}{7} c = {}&& 1 + 2 &&{} + 3 + 4 &&{} + 5 + 6 + \cdots \\ 4c = {}&& 4 &&{} + 8 &&{} + 12 + \cdots \\ c - 4c = {}&& 1 - 2 &&{} + 3 - 4 &&{} + 5 - 6 + \cdots \end{alignat} The second key insight is that the alternating series is the formal
power series expansion (for
x at point 0) of the function which is evaluated with
x defined as 1. Accordingly, Ramanujan writes -3c = 1 - 2 + 3 - 4 + \cdots = \frac{1}{(1 + 1)^2} = \frac14. Dividing both sides by −3, one gets
c = . Generally speaking, it is incorrect to manipulate infinite series as if they were finite sums. For example, if zeroes are inserted into arbitrary positions of a divergent series, it is possible to arrive at results that are not self-consistent, let alone consistent with other methods. In particular, the step is not justified by the
additive identity law alone. For an extreme example, prepending a single zero to the front of the series can lead to a different result. In the series , each term
n is just a number. If the term
n is promoted to a function
n−s, where
s is a complex variable, then one can ensure that only like terms are added. The resulting series may be manipulated in a more rigorous fashion, and the variable
s can be set to −1 later. The implementation of this strategy is called
zeta function regularization.
Zeta function regularization In
zeta function regularization, the series \sum_{n=1}^\infty n is replaced by the series \sum_{n=1}^\infty n^{-s}. The latter series is an example of a
Dirichlet series. When the real part of is greater than 1, the Dirichlet series converges, and its sum is the
Riemann zeta function . On the other hand, the Dirichlet series diverges when the real part of is less than or equal to 1, so, in particular, the series that results from setting does not converge. The benefit of introducing the Riemann zeta function is that it can be defined for other values of by
analytic continuation. One can then define the zeta-regularized sum of to be . From this point, there are a few ways to prove that . One method, along the lines of Euler's reasoning, uses the relationship between the Riemann zeta function and the
Dirichlet eta function . The eta function is defined by an alternating Dirichlet series, so this method parallels the earlier heuristics. Where both Dirichlet series converge, one has the identities: \begin{alignat}{7} \zeta(s)&{}={}&1^{-s}+2^{-s}&&{}+3^{-s}+4^{-s}&&{}+5^{-s}+6^{-s}+\cdots& \\ 2\times2^{-s}\zeta(s)&{}={}& 2\times2^{-s}&& {}+2\times4^{-s}&&{} +2\times6^{-s}+\cdots& \\ \left(1-2^{1-s}\right)\zeta(s)&{}={}&1^{-s}-2^{-s}&&{}+3^{-s}-4^{-s}&&{}+5^{-s}-6^{-s}+\cdots&=\eta(s). \end{alignat} The identity (1 - 2^{1-s}) \zeta(s) = \eta(s) continues to hold when both functions are extended by analytic continuation to include values of for which the above series diverge. Substituting , one gets . Now, computing is an easier task, as the eta function is equal to the Abel sum of its defining series, which is a
one-sided limit: -3\zeta(-1) = \eta(-1) = \lim_{x\to 1^-} \left(1 - 2x + 3x^2 - 4x^3 + \cdots\right) = \lim_{x\to 1^-}\frac{1}{(1 + x)^2} = \frac14. Dividing both sides by −3, one gets .
Cutoff regularization The method of regularization using a
cutoff function can "smooth" the series to arrive at . Smoothing is a conceptual bridge between zeta function regularization, with its reliance on
complex analysis, and Ramanujan summation, with its shortcut to the
Euler–Maclaurin formula. Instead, the method operates directly on conservative transformations of the series, using methods from
real analysis. The idea is to replace the ill-behaved discrete series \textstyle\sum_{n=0}^N n with a smoothed version \sum_{n=0}^\infty nf\left(\frac{n}{N}\right), where
f is a cutoff function with appropriate properties. The cutoff function must be normalized to ; this is a different normalization from the one used in differential equations. The cutoff function should have enough bounded derivatives to smooth out the wrinkles in the series, and it should decay to 0 faster than the series grows. For convenience, one may require that is
smooth,
bounded, and
compactly supported. One can then prove that this smoothed sum is
asymptotic to , where is a constant that depends on . The constant term of the asymptotic expansion does not depend on : it is necessarily the same value given by analytic continuation, . c = -\frac16 \times \frac{1}{2!} = -\frac{1}{12}. To avoid inconsistencies, the modern theory of Ramanujan summation requires that is "regular" in the sense that the higher-order derivatives of decay quickly enough for the remainder terms in the Euler–Maclaurin formula to tend to 0. Ramanujan tacitly assumed this property. == Physics ==