Convergence of Some Divergent Series! T. Muthukumar tmk@iitk.ac.in 9 Jun 04 The topic of this article, the idea of attaching a finite value to divergent series, is no longer a purely mathematical exercise. These finite values of divergent series have found application in string theory and quantum field theory (Casimir effect. The finite sum of real/complex numbers is always finite. The infinite sum of real/complex numbers can be either finite or infinite. For instance, n(n + k = + +... + n = is finite and k = + + 3 +... = +. If an infinite sum has finite value it is said to converge, otherwise it is said to diverge. Divergence do not always mean it grows to ±. For instance, k, k and ( k are diverging, while k s s C and R(s >, is converging. The convergence of a series is defined by the convergence of its partial sum. For instance, k
diverges because its partial sum s n := k = n(n + diverges, as n. The geometric series z k = z, ( for all z <, because its partial sum s n := zn z n z. Note that the map T (z = is well-defined between T : C \ {} C. z In fact, T is a holomorphic (complex differentiable functions and, hence, analytic. The analytic function T exists for all complex numbers except z = and, inside the unit disk {z C z < }, T is same as the power series in (. Thus, T may be seen as an analytic continuation of outside the unit disk, except at z =. A famous unique continuation result from complex analysis says that the analytic continuation is unique. Thus, one may consider the value of T (z, outside the unit disk, as an extension of the divergent series z k. In this sense, setting z = in T (z, the divergent series may be seen as taking the finite value k = + + 4 + 8 + 6 +... =. Similarly, setting z = in T (z, the oscillating divergent series may be seen as taking the finite value ( k = ( =. z k
This sum coincides with the Cesàro sum which is a special kind of convergence for series which do not diverge to ±. For instance, k diverges in Cesàro sum too. The Cesàro sum of a series is defined as the Observe that the sequence n lim n n i= ( i a k ( i a k. i= ( k. Since T is not defined on, we have not assigned any finite value to the divergent series + + +.... The Riemann zeta function ζ(z := converges for z C with R(z >. Note that for z C, = e z ln(k, where ln is the real logarithm. If z = σ + it then = k σ e it ln(k and = k σ since e it ln(n =. Therefore, the infinite series converges for all z C such that R(z > (i.e. σ > because it is known that k σ converges for all σ > and diverges for all σ. The Riemann zeta function is a special case of the Dirichlet series D(z := a k 3
with a k = for all k. Note that for z =, we get the Basel series ζ( := k = π 6 and ζ(3 =.005690359594... called the Apéry s constant. In fact, the following result of Euler gives the value of Riemann zeta function for all even positive integers. Theorem (Euler. For all k N, k+ (πk ζ(k = ( (k! B k where B k is the k-th Bernoulli number. If ζ(z is well-defined for R(z > then z ζ(z is also well-defined for R(z >. Therefore, ζ(z( z = ( k n z =: η(z. Observe that η(z is also a Dirichlet series. It can be shown that ( z η(z is analytic for all R(z > 0 and R(z. Thus, we have analytically continued ζ(z for all R(z > 0 and R(z. There is a little work to be done on the zeroes of z but is fixable. In the strip 0 < R(z <, called the critical strip, the Riemann zeta function satisfies the relation ( zπ ζ(z = z (π z sin Γ( zζ( z. This relation is used to extend the Riemann zeta function to z with nonpositive real part, thus, extending to all complex number z. Setting z = in the relation yields ζ( = π sin ( π Since ζ( is extension of the series Γ(ζ( = π ( π 6 =. 4
one may think of ζ( as the finite value corresponding to + + 3 +.... The value of ζ(0 is obtained limiting process because it involves ζ( which is not defined. However, there is an equivalent formulation of Riemann zeta function for all z C \ {} as ζ(z = z k+ k ( k ( i (i + z i with a simple pole and residue at z =. Recall that the analytic continuation T of the geometric series is not defined for z = and, hence, we could not assign a finite value to But, setting z = 0 above, T ( = ζ(0 = i=0 T ( = + + +.... k+ k ( k ( i δ 0k = i = k+ i=0 where δ 0k is the Kronecker delta defined as { k = 0 δ 0k = 0 k 0. The harmonic series n= corresponds to ζ( which is not defined. The closest one can conclude about ζ( is that ( lim n k ln(n = γ n where γ is the Euler-Mascheroni constant which has the value γ = 0.577566.... 5