Divergent Series Past, Present, Future

Divergent series past, present, future. . .

Christiane Rousseau

1 Divergent series, JMM 2015 How many students and mathematicians have never heard of divergent series?

2 Divergent series, JMM 2015 Preamble: My presentation of the subject and of some of its history is not exhaustive. Moreover, it is biased by my interest in dynamical systems.

3 Divergent series, JMM 2015 Euler:

“If we get the sum

1 − 1 + 1 − 1 + 1 − 1 + ···

1 its only reasonable value is 2.”

Divergent series have been used a lot in the past by people including Fourier, Stieltjes, Euler, . . .

4 Divergent series, JMM 2015 Divergent series have been used a lot in the past by people including Fourier, Stieltjes, Euler, . . .

Euler:

“If we get the sum

1 − 1 + 1 − 1 + 1 − 1 + ···

1 its only reasonable value is 2.”

5 Divergent series, JMM 2015 A hundred years before Riemann, Euler had found the functional equation of the function ζ in the form

1s−1 − 2s−1 + 3s−1 − ··· (s − 1)!(2s − 1) 1 = − cos sπ 1 1 1 (2s−1 − 1)πs 2 1s − 2s + 3s − ···

by “calculating”, for s ∈ N ∪ {1/2,3/2}, the sums

1s − 2s + 3s − 4s + 5s − 6s + ··· and 1 1 1 1 1 1 − + − + − + ··· 1s 2s 3s 4s 5s 6s

6 Divergent series, JMM 2015 Where is the turn?

7 Divergent series, JMM 2015 Cauchy, Abel

8 Divergent series, JMM 2015 Cauchy made one exception: he justiﬁed rigorously the use of Stirling’s divergent series in numerical computations.

Cauchy, Preface of “Analyse mathematique”,´ 1821

“I have been forced to admit some propositions which will seem, perhaps, hard to accept. For instance, that a divergent series has no sum.”

9 Divergent series, JMM 2015 Cauchy, Preface of “Analyse mathematique”,´ 1821

“I have been forced to admit some propositions which will seem, perhaps, hard to accept. For instance, that a divergent series has no sum.” Cauchy made one exception: he justiﬁed rigorously the use of Stirling’s divergent series in numerical computations.

10 Divergent series, JMM 2015 We can prove anything by using them and they have caused so much misery and created so many paradoxes. . . ”

Niels Henrik Abel, letter to Holmboe (1826)

“Divergent series are, in general, something terrible, and it is a shame to base any proof on them.

11 Divergent series, JMM 2015 Niels Henrik Abel, letter to Holmboe (1826)

“Divergent series are, in general, something terrible, and it is a shame to base any proof on them. We can prove anything by using them and they have caused so much misery and created so many paradoxes. . . ”

12 Divergent series, JMM 2015 The geometers, concerned with absolute rigor and not bothered by the length of the inextricable computations that they conceive to be possible without trying to undertake them explicitly, would say that a series is convergent when the sum of the terms tends to a definite limit, even if the first terms decrease very slowly. On the contrary, the astronomers use to say that a series converges when, for instance, the first 20 terms decrease very rapidly, even if the remaining terms would grow forever.

Poincare,´ second volume of the New Methods of Celestial Mechanics (1893)

“There is a kind of misunderstanding between the geometers and the astronomers, concerning the meaning of the word convergence.

13 Divergent series, JMM 2015 On the contrary, the astronomers use to say that a series converges when, for instance, the ﬁrst 20 terms decrease very rapidly, even if the remaining terms would grow forever.

Poincare,´ second volume of the New Methods of Celestial Mechanics (1893)

“There is a kind of misunderstanding between the geometers and the astronomers, concerning the meaning of the word convergence. The geometers, concerned with absolute rigor and not bothered by the length of the inextricable computations that they conceive to be possible without trying to undertake them explicitly, would say that a series is convergent when the sum of the terms tends to a deﬁnite limit, even if the ﬁrst terms decrease very slowly.

14 Divergent series, JMM 2015 Poincare,´ second volume of the New Methods of Celestial Mechanics (1893)

15 Divergent series, JMM 2015 The geometers will say that the first series converges, and even that it converges fast . . . ; and they will say that the second series diverges. . . On the contrary, the astronomers will consider the first series as divergent, . . . , and the second series as convergent. The two rules are legitimate: the first one in the theoretical researches; the second one in the numerical applications . . . ”

Thus, let us take a simple example and consider the two series which have as general term 1000n n! and . n! 1000n

16 Divergent series, JMM 2015 On the contrary, the astronomers will consider the ﬁrst series as divergent, . . . , and the second series as convergent. The two rules are legitimate: the ﬁrst one in the theoretical researches; the second one in the numerical applications . . . ”

Thus, let us take a simple example and consider the two series which have as general term 1000n n! and . n! 1000n

The geometers will say that the ﬁrst series converges, and even that it converges fast . . . ; and they will say that the second series diverges. . .

17 Divergent series, JMM 2015 The two rules are legitimate: the ﬁrst one in the theoretical researches; the second one in the numerical applications . . . ”

Thus, let us take a simple example and consider the two series which have as general term 1000n n! and . n! 1000n

18 Divergent series, JMM 2015 Thus, let us take a simple example and consider the two series which have as general term 1000n n! and . n! 1000n

The geometers will say that the first series converges, and even that it converges fast . . . ; and they will say that the second series diverges. . . On the contrary, the astronomers will consider the first series as divergent, . . . , and the second series as convergent. The two rules are legitimate: the first one in the theoretical researches; the second one in the numerical applications . . . ”

19 Divergent series, JMM 2015 For instance if a power series is a solution of a differential equation, then its sum should be a function which is a solution of this differential equation.

Divergent series for solving real problems

Emile´ Borel: a technique for assigning a sum to a series should be adapted to solve a mathematical problem.

20 Divergent series, JMM 2015 Divergent series for solving real problems

Emile´ Borel: a technique for assigning a sum to a series should be adapted to solve a mathematical problem. For instance if a power series is a solution of a differential equation, then its sum should be a function which is a solution of this differential equation.

21 Divergent series, JMM 2015 Hence, ^f (x) = (−1)nn! xn+1 ∞= Xn 0

Euler’s differential equation

x2y0 + y − x = 0 This equation has the formal solution

^ n+1 f (x) = anx ∞= Xn 0 where a0 = 1 an = −nan−1

22 Divergent series, JMM 2015 Euler’s differential equation

x2y0 + y − x = 0 This equation has the formal solution

^ n+1 f (x) = anx ∞= Xn 0 where a0 = 1 an = −nan−1 Hence, ^f (x) = (−1)nn! xn+1 ∞ n=0 23 X Divergent series, JMM 2015 Borel’s vision A convergent series

n+1 anx ∞ = Xn 0 has a disk of convergence B(0,r). There exists at least one singularity on the boundary of the disk of convergence

24 Divergent series, JMM 2015 We can extend the function by turning around the singularities. Generically this extension is ramiﬁed.

5πi log(1 − x) + log(i − x) + log e 4 − x

25 Divergent series, JMM 2015 In Borel’s mind, a divergent series is a convergent series on a disk of radius r = 0. If there are only a finite number of singularities on the boundary of the “disk”, i.e. in a finite number of directions, then we can extend the function around these, and we get a function defined on sectors.

26 Divergent series, JMM 2015 In Borel’s mind, a divergent series is a convergent series on a disk of radius r = 0. If there are only a finite number of singularities on the boundary of the “disk”, i.e. in a finite number of directions, then we can extend the function around these, and we get a function defined on sectors.

27 Divergent series, JMM 2015 To ﬁnd them we “blow-up”

Where are hidden these singularities?

28 Divergent series, JMM 2015 Where are hidden these singularities?

To ﬁnd them we “blow-up”

29 Divergent series, JMM 2015 I If S = n=0 an, then P∞ S − a0 = an ∞ Xn=1 0 I If S = n=0 an and S = n=0 bn, then ∞ ∞ P 0 P S + cS = (an + cbn) ∞= Xn 0

Basic rules for a summation method

I If n=0 an is convergent, then the method should provide the usual sum. P∞

30 Divergent series, JMM 2015 0 I If S = n=0 an and S = n=0 bn, then ∞ ∞ P 0 P S + cS = (an + cbn) ∞= Xn 0

Basic rules for a summation method

I If n=0 an is convergent, then the method should provide the usual sum. P∞ I If S = n=0 an, then P∞ S − a0 = an ∞ Xn=1

31 Divergent series, JMM 2015 Basic rules for a summation method

I If n=0 an is convergent, then the method should provide the usual sum. P∞ I If S = n=0 an, then P∞ S − a0 = an ∞ Xn=1 0 I If S = n=0 an and S = n=0 bn, then ∞ ∞ P 0 P S + cS = (an + cbn) ∞= Xn 0

32 Divergent series, JMM 2015 0 I If S = n=0 an and S = n=0 bn are absolutely summable, then the product P∞ P∞ series is absolutely summable   0 S · S =  ajbk ∞= Xn 0 j+Xk=n

33 Divergent series, JMM 2015 Borel summation method of divergent power series

n+1 Let us take a convergent series n=0 anx P∞ n+1 n+1 anx anx = n! ∞ ∞ n! = = Xn 0 Xn 0 a xn+1 = yne−ydy n ∞ n! = 0∞ Xn 0 Z n! a (xy)n = e|−y {zn } xdy ζ = xy ∞ n! 0∞ = Z Xn 0 dζ n − ζ anζ |{z} = e x dζ ∞ n! 0∞ = Z Xn 0

34 Divergent series, JMM 2015 The Borel sum of a series

This suggests deﬁning

n − ζ anζ e x dζ n! 0∞ ∞= Z Xn 0 n+1 as the sum of the series n=0 anx , whenever this expression makes sense. P∞

35 Divergent series, JMM 2015 Summability on the direction d n+1 The power series n=0 anx is Borel-summable in the directiond if P∞ I a ζn n dζ ∞ n! = Xn 0 is convergent in a disk B(0,r), I the sum of this series can be extended along the half-line d and its growth is at most exponential at inﬁnity. Then the sum of the series is given by

n n+1 − ζ anζ a x = e x dζ n n! ∞ d ∞= X0 Z Xn 0

36 Divergent series, JMM 2015 Borel summability

n+1 The power series n=0 anx is Borel-summable if it is Borel-summable in the direction d for all directions d, but aP ﬁnite∞ number.

37 Divergent series, JMM 2015 The modern presentation of this process

Three steps:

I We “blow-up”to send the hidden singularities at the singular point to a ﬁnite distance: this is done by the Borel transform.

I We extend along half-lines avoiding the singularities.

I We compute the sum in applying the Laplace transform.

38 Divergent series, JMM 2015 Borel transform

an ^f (x) = a xn+1 7 B(^f )(ζ) = ζn n n! ∞= ∞= Xn 0 Xn 0 →

39 Divergent series, JMM 2015 Extending along half-lines avoiding the singularities

40 Divergent series, JMM 2015 Laplace transform

− ζ B(^f )(ζ) 7 L B(^f ) (x) = e x B(^f )(ζ) dζ Zd →

41 Divergent series, JMM 2015 Coming back to Euler’s equation x2y0 + y − x = 0

^ n n+1 The formal solution is f (x) = n=0(−1) n! x . Hence ∞ P 1 B(^f )(ζ) = (−1)nζn = ∞ 1 + ζ = Xn 0 and − ζ e x L B(^f ) (x) = dζ 1 + ζ Zd

which is a solution of the differential equation on the domain

42 Divergent series, JMM 2015 Comparison with the solution obtained by variation of the constant

The solutions of the linear system are given by x 1 1 − 1 f (x) = e x e ξ dξ + C ξ Z0 For x > 0, the solution which is asymptotic to0 corresponds to C = 0. 1 1 ζ The change of variable x − ξ = − x provides the solution given by Borel’s summation method

− ζ e x f (x) = dζ 1 + ζ Z0∞

43 Divergent series, JMM 2015 Theorem (Borel) If

I P is a multivariate polynomial and, ^ I y = f (x) is a formal solution of the differential equation

P(x,y,y0,...,y(n)) = 0 (∗)

^ I and if f (x) is absolutely Borel-summable with sum f (x), then the function y = f (x) is solution of the differential equation (*).

A summation method is useful if we have a theorem

An example of such a theorem

44 Divergent series, JMM 2015 ^ I y = f (x) is a formal solution of the differential equation

P(x,y,y0,...,y(n)) = 0 (∗)

^ I and if f (x) is absolutely Borel-summable with sum f (x), then the function y = f (x) is solution of the differential equation (*).

A summation method is useful if we have a theorem

An example of such a theorem

Theorem (Borel) If

I P is a multivariate polynomial and,

45 Divergent series, JMM 2015 ^ I and if f (x) is absolutely Borel-summable with sum f (x), then the function y = f (x) is solution of the differential equation (*).

A summation method is useful if we have a theorem

An example of such a theorem

Theorem (Borel) If

I P is a multivariate polynomial and, ^ I y = f (x) is a formal solution of the differential equation

P(x,y,y0,...,y(n)) = 0 (∗)

46 Divergent series, JMM 2015 then the function y = f (x) is solution of the differential equation (*).

A summation method is useful if we have a theorem

An example of such a theorem

Theorem (Borel) If

I P is a multivariate polynomial and, ^ I y = f (x) is a formal solution of the differential equation

P(x,y,y0,...,y(n)) = 0 (∗)

^ I and if f (x) is absolutely Borel-summable with sum f (x),

47 Divergent series, JMM 2015 A summation method is useful if we have a theorem

An example of such a theorem

Theorem (Borel) If

I P is a multivariate polynomial and, ^ I y = f (x) is a formal solution of the differential equation

P(x,y,y0,...,y(n)) = 0 (∗)

^ I and if f (x) is absolutely Borel-summable with sum f (x), then the function y = f (x) is solution of the differential equation (*).

48 Divergent series, JMM 2015 Approximation of the function by a partial sum

Let us show that, if we approximate the solution

− ζ e x f (x) = dζ 1 + ζ Z0∞ by a partial sum

n k k+1 Sn(x) = (−1) k! x Xk=0 then, for x > 0, the error is smaller than the ﬁrst neglected term n+2 |f (x) − Sn(x)| ≤ (n + 1)! x

49 Divergent series, JMM 2015 Details

n k k+1 Sn(x) = (−1) k! x k=0 Xn = (−1)kxk+1 e−yyk dy 0∞ Xk=0 Z n ζ = (−1)kxk+1e−yyk dyy = x Z0∞ k=0 Xn k − ζ k = (−1) e x ζ dζ 0∞ Z Xk=0

50 Divergent series, JMM 2015 Details

Then n − ζ 1 x k k |f (x) − Sn(x)| ≤ e − (−1) ζ dζ 0∞ 1 + ζ Z Xk=0

(−1)n+1ζn+1

1+ζ | {z }

51 Divergent series, JMM 2015 Details (end)

n+1 n+1 − ζ (−1) ζ |f (x) − Sn(x)| ≤ e x dζ 1 + ζ Z0∞ − ζ n+ ≤ e x ζ 1dζζ = xy Z0∞ = e−yxn+2yn+1 dy Z0∞ = (n + 1)! xn+2

52 Divergent series, JMM 2015 What is the size of the rest? Of the order of √ n+2 −(n+1) n+ 1 1 (n + 1)! x ∼ (n + 1) 2π e (n + 1) 2 (n + 1)n+2 √ √ − 1 ∼ 2π x e x The approximation is exponentially precise!

Quality of the approximation

We must choose n so that (n + 1)! xn+2 be the smallest 1 possible, namely n + 1 ∼ x

53 Divergent series, JMM 2015 Of the order of √ n+2 −(n+1) n+ 1 1 (n + 1)! x ∼ (n + 1) 2π e (n + 1) 2 (n + 1)n+2 √ √ − 1 ∼ 2π x e x The approximation is exponentially precise!

Quality of the approximation

We must choose n so that (n + 1)! xn+2 be the smallest 1 possible, namely n + 1 ∼ x What is the size of the rest?

54 Divergent series, JMM 2015 The approximation is exponentially precise!

Quality of the approximation

We must choose n so that (n + 1)! xn+2 be the smallest 1 possible, namely n + 1 ∼ x What is the size of the rest? Of the order of √ n+2 −(n+1) n+ 1 1 (n + 1)! x ∼ (n + 1) 2π e (n + 1) 2 (n + 1)n+2 √ √ − 1 ∼ 2π x e x

55 Divergent series, JMM 2015 Quality of the approximation

56 Divergent series, JMM 2015 Quality of the approximation

This example is far from an exception. Poincare´ would have said that this series is “convergent for the astronomers”: “On the contrary, the astronomers use to say that a series converges when, for instance, the ﬁrst 20 terms decrease very rapidly, even if the remaining terms would grow forever.”

57 Divergent series, JMM 2015 ...

For the most part, it is true that the results are correct, which is very strange. I am working to ﬁnd out why, a very interesting problem.”

The present

Coming back to Abel’s citation

“Divergent series are, in general, something terrible and it is a shame to base any proof on them. We can prove anything by using them and they have caused so much misery and created so many paradoxes

58 Divergent series, JMM 2015 The present

Coming back to Abel’s citation

...

For the most part, it is true that the results are correct, which is very strange. I am working to ﬁnd out why, a very interesting problem.”

59 Divergent series, JMM 2015 Coming back to Abel’s citation

...

For the most part, it is true that the results are correct, which is very strange. I am working to ﬁnd out why, a very interesting problem.”

The present

60 Divergent series, JMM 2015 Finally my eyes were suddenly opened since, with the exception of the simplest cases, for instance the geometric series, we hardly ﬁnd, in mathematics, any inﬁnite series whose sum may be determined in a rigorous fashion,

which means the most essential part of mathematics has no foundation.

For the most part, it is true that the results are correct, which is very strange. I am working to ﬁnd out why, a very interesting problem.”

Abel’s citation

61 Divergent series, JMM 2015 which means the most essential part of mathematics has no foundation.

For the most part, it is true that the results are correct, which is very strange. I am working to ﬁnd out why, a very interesting problem.”

Abel’s citation

Finally my eyes were suddenly opened since, with the exception of the simplest cases, for instance the geometric series, we hardly ﬁnd, in mathematics, any inﬁnite series whose sum may be determined in a rigorous fashion,

62 Divergent series, JMM 2015 For the most part, it is true that the results are correct, which is very strange. I am working to ﬁnd out why, a very interesting problem.”

Abel’s citation

which means the most essential part of mathematics has no foundation.

63 Divergent series, JMM 2015 Abel’s citation

which means the most essential part of mathematics has no foundation.

For the most part, it is true that the results are correct, which is very strange. I am working to ﬁnd out why, a very interesting problem.”

64 Divergent series, JMM 2015 The future

What does Abel mean when he says that divergent series appear in the most essential part of mathematics?

65 Divergent series, JMM 2015 Why?

There are indeed in mathematics many situations where divergence is the rule and convergence the exception.

66 Divergent series, JMM 2015 There are indeed in mathematics many situations where divergence is the rule and convergence the exception.

Why?

67 Divergent series, JMM 2015 The center manifold is given by Euler’s equation x2y0 + y − x = 0

The center manifold of a saddle node

The center manifold of a saddle-node of a 2-dimensional analytic vector ﬁeld is almost never analytic.

x˙ = x2 y˙ = −y + x

68 Divergent series, JMM 2015 The center manifold of a saddle node

The center manifold of a saddle-node of a 2-dimensional analytic vector ﬁeld is almost never analytic.

x˙ = x2 y˙ = −y + x

The center manifold is given by Euler’s equation x2y0 + y − x = 0

69 Divergent series, JMM 2015 To understand we complexify x,y ∈ C

x˙ = x2 y˙ = −y + x

70 Divergent series, JMM 2015 Because there is a hidden singularity in one direction. . . To understand we unfold

x˙ = x2 − y˙ = −y + x

Why is the center manifold generically ramiﬁed at a saddle-node?

71 Divergent series, JMM 2015 To understand we unfold

x˙ = x2 − y˙ = −y + x

Why is the center manifold generically ramiﬁed at a saddle-node?

Because there is a hidden singularity in one direction. . .

72 Divergent series, JMM 2015 Why is the center manifold generically ramiﬁed at a saddle-node?

Because there is a hidden singularity in one direction. . . To understand we unfold

x˙ = x2 − y˙ = −y + x

73 Divergent series, JMM 2015 Indeed, the saddle has an analytic invariant manifold. At the limit the node becomes the hidden singularity.

74 Divergent series, JMM 2015 √ Normal form at a node if2 ∈ / −1/N

√ x˙ = −2 x y˙ = −y

All solutions are of the form

− √1 y = Cx 2 . They are all ram- iﬁed except the one for C = 0

But why should we have ramiﬁcation at the node generically?

75 Divergent series, JMM 2015 But why should we have ramiﬁcation at the node generically?

√ Normal form at a node if2 ∈ / −1/N

√ x˙ = −2 x y˙ = −y

All solutions are of the form

− √1 y = Cx 2 . They are all ram- iﬁed except the one for C = 0

76 Divergent series, JMM 2015 Non resonant case

The generic situation is that the analytic invariant mani- folds of the saddle and the node do not match.

77 Divergent series, JMM 2015 This is necessarily the case for small when unfolding a saddle-node with non analytic center manifold.

78 Divergent series, JMM 2015 Case of a resonant node

√ Normal form at a node if2 ∈ −1/N 1 x˙ = − x n y˙ = −y + axn

The solutions are the form

y = Cxn − axn logx. They are all ramiﬁed if a 6= 0, and none is ram- iﬁed if a = 0.

79 Divergent series, JMM 2015 Parametric resurgence phenomenon in the resonant case

When unfolding a saddle- node with non analytic center manifold then, for small , the node is necessarily non linearizable when resonant.

80 Divergent series, JMM 2015 Parametric resurgence phenomenon in the resonant case

When unfolding a saddle- node with non analytic center manifold then, for small , the node is necessarily non linearizable when resonant.

81 Divergent series, JMM 2015 The phenomenon described above is quite generic and geometric explanations in this spirit are valid in many examples.

Conclusion

We have understood why divergence is the rule, and convergence, the exception.

82 Divergent series, JMM 2015 Conclusion

We have understood why divergence is the rule, and convergence, the exception.

The phenomenon described above is quite generic and geometric explanations in this spirit are valid in many examples.

83 Divergent series, JMM 2015 In dynamical systems

I Borel-summability (also called1 -summability) occurs when 2 special objects (singular points, limit cycles, etc.) coallesce.

I k-summability or multisummability occurs when k + 1 special objects coallesce.

I Divergence could occur through small divisors: an explanation was proposed by Yoccoz and further studied by Perez-Marco.

84 Divergent series, JMM 2015 An example studied with Colin Christopher

The saddle point in x˙ = x(1 − x + y) y˙ = y(−λ + x + dy) 1 is linearizable for all values of d, except when λ = 1 + n , in which case it is integrable only for the values of d given by the dots.

–1

–1.5

–2

–2.5

–3

–3.5

1.1 1.2 1.3 1.4 1.5

85 Divergent series, JMM 2015 The end

86 Divergent series, JMM 2015