•construct a rearrangement of the MRB terms, (-1)^n (n^(1/n)-1) for n element of N, that converges to e, pi, and the golden ratio

Let

a

n



n

(-1)

(

1/n

n

-1), n∈N.

You are asking for rearrangements of the terms of the series

∞

∑

n1

a

n

that converge to prescribed values such as

e

,

π

, and

φ

.

Key fact

Since

1/n

n

-1

(logn)/n

e

-1∼

logn

n

,

we have

∞

∑

n1

(

1/n

n

-1)∞,

while the alternating series

∞

∑

n1

n

(-1)

(

1/n

n

-1)

converges by the alternating test, but not absolutely. Therefore, by the Riemann rearrangement phenomenon, one can rearrange the terms to make the series converge to any real number. So in particular, there exist rearrangements converging to

e

,

π

, and

φ

1+

5

2

.

A constructive rearrangement

Split the terms into positive and negative parts:

• positive terms occur for even

n

:

p

k



1/(2k)

(2k)

-1,

• negative terms occur for odd

n

:

q

k

-

1/(2k-1)

(2k-1)

-1.

Then both

∞

∑

k1

p

k

∞,

∞

∑

k1

q

k

-∞.

To construct a rearrangement converging to a target

t

:1. Add unused positive terms until the partial sum first exceeds

t

.2. Then add unused negative terms until the partial sum first drops below

t

.3. Repeat forever.Because the term sizes go to 0, the oscillation around

t

shrinks to 0, so the rearranged series converges to

t

.

Wolfram Language construction

Here is a simple constructor for the first m terms of such a rearrangement toward a target target:

clearRearrangement[target_,m_]:=Module[{evenIndex=1,oddIndex=1,sum=0,terms={},nextPositive,nextNegative},nextPositive[]:=((2evenIndex)^(1/(2evenIndex))-1);nextNegative[]:=-(((2oddIndex-1)^(1/(2oddIndex-1)))-1);While[Length[terms]<m,While[sum≤target&&Length[terms]<m,AppendTo[terms,{2evenIndex,nextPositive[]}];sum+=nextPositive[];evenIndex++;];While[sum≥target&&Length[terms]<m,AppendTo[terms,{2oddIndex-1,nextNegative[]}];sum+=nextNegative[];oddIndex++;];];terms]

Insert and evaluate

Insert

Copy

A cleaner functional version returning just the rearranged indices is:

rearrangedIndices[target_,m_]:=Module[{evenIndex=1,oddIndex=1,sum=0.,indices={},pos,neg},pos[k_]:=(2k)^(1/(2k))-1;neg[k_]:=-((2k-1)^(1/(2k-1))-1);While[Length[indices]<m,While[sum≤target&&Length[indices]<m,sum+=pos[evenIndex];AppendTo[indices,2evenIndex];evenIndex++;];While[sum≥target&&Length[indices]<m,sum+=neg[oddIndex];AppendTo[indices,2oddIndex-1];oddIndex++;];];indices]

Insert and evaluate

Insert

Copy

Then define the rearranged partial sums:

a[n_]:=(-1)^n(n^(1/n)-1);rearrangedPartialSums[target_,m_]:=Module[{idx=rearrangedIndices[target,m]},Accumulate[a/@idx]]

Insert and evaluate

Insert

Copy

For the three requested targets

Use:

eIndices=rearrangedIndices[E,2000];piIndices=rearrangedIndices[Pi,2000];phiIndices=rearrangedIndices[GoldenRatio,2000];

Insert and evaluate

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Copy

and partial sums:

eSums=Accumulate[a/@eIndices];piSums=Accumulate[a/@piIndices];phiSums=Accumulate[a/@phiIndices];

Insert and evaluate

Insert

Copy

You should see:

Last[eSums]Last[piSums]Last[phiSums]

Insert and evaluate

Insert

Copy

approach respectively:

EPiGoldenRatio

Insert and evaluate

Insert

Copy

More explicit mathematical description

If you want a purely mathematical definition, for a target

t∈R

define inductively:

• choose the smallest unused even indices until the sum exceeds

t

;

• then choose the smallest unused odd indices until the sum is below

t

;

• repeat.

This gives a permutation

σ

of

N

such that

∞

∑

k1

σ(k)

(-1)

σ

1/σ(k)

(k)

-1t.

So there are permutations

σ

e

,

σ

π

,

σ

φ

with

∞

∑

k1

σ

e

(k)

(-1)



σ

e

1/

σ

e

(k)

-1e,

∞

∑

k1

σ

π

(k)

(-1)



σ

π

1/

σ

π

(k)

-1π,

∞

∑

k1

σ

φ

(k)

(-1)



σ

φ

1

σ

φ

(k)

-1φ.

Small note

Your terms start at

n∈N

. If your convention includes 0, then

1/0

0

is undefined, so this sequence should start at

n1

.If you want, I can also give you:1. a version that returns the rearrangement as an actual permutation, 2. a proof that this greedy construction converges to the target, or 3. a notebook that plots the three rearranged partial sums approaching

e

,

π

, and

φ

.

a notebook that plots the three rearranged partial sums

Rearranged partial sums of the MRB series

Definitions

Compute the three rearrangements

Plot each one separately

Combined plot

Final values and errors

Optional: show the first few selected indices