Numerical evaluation of a sum

I am trying to compute numerically NSum[(-1)^n/n^3, {n, 1, Infinity}]. Of course, using first Sum would work here, but often it's not an option (no closed-form). I would like to compute the sum with an arbitrary number of digits.

So far, I've found that

NSum[(-1)^n/n^3, {n, 1, Infinity},
WorkingPrecision -> 24,
AccuracyGoal -> 24, PrecisionGoal -> Infinity]


will give only 16 digits (-0.9015426773696957), whatever the value of WorkingPrecision and AccuracyGoal.

However, if I add the option Method -> "AlternatingSigns", I get as many digits as I want, and it's really controlled by WorkingPrecision.

Among the other options, Method -> "EulerMaclaurin" gives a wrong answer. Not all digits are correct, and there is an imaginary part: -0.901542677369695714049804 + 15.503138340149910087738157 I. And Method -> "WynnEpsilon" also fails, giving -0.9015426773696957140, even if I ask for more digits.

So, is there something I'm doing wrong? Is there a reason it does not work if I don't give a specific method? Is there a more straightforward way to compute a sum (or any numerical result, for that matter)?

And, since I'm quite interested by numerical computations with Mathematica and I'm a beginner (with Mathematica, not with numerical analysis :-)), is there a good guide out there ?

As an aside, I know Maxima better, and I'm used to the option fpprec, that controls "bigfloat" evaluation in every computations. Is there something similar in Mathematica? I think it has a better control of accuracy, so maybe the question does not make sense (maybe there is no "global" setting).

To answer ciao's comment, since there is not much space in comments:

Actually, if I use larger values of the option NSumTerms, I get more and more digits, but they don't agree with what I get with Method -> "AlternatingSigns":

NSum[(-1)^n/n^3, {n, 1, Infinity}, WorkingPrecision -> 50,
AccuracyGoal -> 50, PrecisionGoal -> Infinity, NSumTerms -> 5000]


-0.90154267736969571404980362113358749307373971926

NSum[(-1)^n/n^3, {n, 1, Infinity}, WorkingPrecision -> 50,
AccuracyGoal -> 50, PrecisionGoal -> Infinity,
Method -> "AlternatingSigns"]


-0.90154267736969378050491347983360150530995076960417

After 14 digits, they are different numbers.

And here it's possible to check:

Sum[(-1)^n/n^3, {n, 1, Infinity}] // N[#, 50] &


-0.90154267736969571404980362113358749307373971925537

So, the answer with Method -> "AlternatingSigns" is also wrong!

And it's not better if I combine Method and NSumTerms options:

NSum[(-1)^n/n^3, {n, 1, Infinity}, WorkingPrecision -> 50,
AccuracyGoal -> 50, PrecisionGoal -> Infinity,
Method -> "AlternatingSigns", NSumTerms -> 5000]


-0.90154267736969571404546474460914681065800480608588

So the results differ after 20 digits.

Now I'm even more puzzled :-) I can see that your comment is right, since I get the right answer, but I absolutely don't understand why, and why the other methods fail without notification.

Also, it's not a very satisfying approach, since I can't set the number of accurate digits in the output (it depends on NSumTerms in a non obvious way).

• NSumTerms is your friend...
– ciao
Jun 7 '14 at 9:00
• @rasher, I edited the post to answer your comment, since I lacked space.
– user10957
Jun 7 '14 at 9:13
• @Jean-ClaudeArbaut - Sum and NSum with the NSumTerms-option deliver the same result (write N[#, 47]&)
– eldo
Jun 7 '14 at 9:30
• @eldo, yes, but I had to choose a large enough value of NSumTerms (1000 is not enough for 50 digits, 5000 is almost enough). Is there a way to simply tell Mathematica "I want 50 digits, choose the options as you wish"? Maybe I'm asking for too much, but I find rather strange that many options give wrong results, and one option give the right answer, but with the correct parameter. Not easy to do that automatically. How am I supposed to know I've found the right option, if I can't check with closed form?
– user10957
Jun 7 '14 at 9:43
• One should always strive to exploit the structure of a problem whenever possible. In this case, since it is apparent that the sum to be evaluated is alternating, you should definitely be using the "AlternatingSigns" method. Feb 14 '16 at 2:38

As a general rule, one should usually set WorkingPrecision to be higher than either of AccuracyGoal and PrecisionGoal (if not set to ∞) to mitigate any roundoff error incurred during the evaluation.

For the specific methods implemented within NSum[]: it is documented, but not apparently well-known, that you can change the way the integral term in the Euler-Maclaurin method is evaluated numerically. For instance,

NSum[(-1)^n/n^3, {n, 1, ∞},
WorkingPrecision -> 30, AccuracyGoal -> 24, PrecisionGoal -> ∞,
Method -> {"EulerMaclaurin",
Method -> {NIntegrate, Method -> "GlobalAdaptive"}}]
-0.901542677369695714049804 + 0.*10^-25 I


yields a result quite close to $\frac34\zeta(3)$.

For the Wynn $\varepsilon$ method, what you can tweak is the degree of the transformation used (the default Degree -> 1 corresponds to the Aitken $\Delta^2$ process) and the number of terms of the sum to be used in the extrapolation, "ExtraTerms". (Recall that in evaluating a sum by extrapolation, one usually splits the sum in two parts, where one part is evaluated through direct summation, and the "tail" is evaluated by using extrapolation.)

Thus,

NSum[(-1)^n/n^3, {n, 1, ∞},
WorkingPrecision -> 30, AccuracyGoal -> 24, PrecisionGoal -> ∞,
NSumTerms -> 20, Method -> {"WynnEpsilon", Degree -> 2, "ExtraTerms" -> 20}]
-0.901542677369695714049803590206


Of course, NSum[] is not always able to return good answers, so anybody who is sufficiently paranoid about his results should always look into a number of other summation methods. But that is for another time for me to expound on...

• As for the "AlternatingSigns" method, this yields a result good to ~ 80 digits: NSum[(-1)^n/n^3, {n, 1, Infinity}, WorkingPrecision -> 80, AccuracyGoal -> 50, PrecisionGoal -> ∞, Method -> {"AlternatingSigns", "ExtraTerms" -> 100}, NSumTerms -> 100]. Feb 15 '16 at 14:35

The author of the question says this in a comment:

Is there a way to simply tell Mathematica "I want 50 digits, choose the options as you wish"? Maybe I'm asking for too much, but I find rather strange that many options give wrong results, and one option give the right answer, but with the correct parameter.

Here are some comments of mine in that regard.

1. The Numerical Summation Extrapolation (NSE) methods seem to have narrow scopes of applicability within the "space" of number sequences. If the model of a NSE method fits the sequence that we work with, then the NSE method is usually very effective. That same method can be failing really badly with sequences that do not fit its model.

2. From that perspective NSum's set of methods is deficient. E.g. NSum does not have Richardson extrapolation.

3. In general Mathematica tries to provide algorithms that select the methods and parameters to provide the answers the users want. So with "I want 50 digits, choose the options as you wish" you are not asking too much from Mathematica (generally speaking) but you are asking too much from NSum.

Code

Here is an implementation that solves the particular problem posted in the question. It uses the Shanks transformation, and a search procedure that is using the difference of consecutive results for a stopping criteria. (As in Cauchy's convergence test.) I tried the Richardson extrapolation too but it did not give good results. (Using the Shanks-vs-Richardson comparison code in this answer of mine.)

First we define the Shanks transformation function:

ClearAll[Shanks]
Shanks[A_, n_] := (A[n + 2] A[n] - A[n + 1]^2)/(A[n + 2] + A[n] - 2 A[n + 1]);
Shanks[A_, 1, n_] := Shanks[A, 1, n] = Shanks[A, n];
Shanks[A_, k_, n_] :=
Shanks[A, k, n] = ((Shanks[A, -1 + k, n] Shanks[A, -1 + k, 2 + n] -
Shanks[A, -1 + k, 1 + n]^2)/(Shanks[A, -1 + k, n] +
Shanks[A, -1 + k, 2 + n] - 2 Shanks[A, -1 + k, 1 + n]));


Here is the search function:

Clear[SearchSumValue]
SearchSumValue[partialSumFunc_, accGoal_, step_: 10, startStep_: 40, methodFunc_: Shanks] :=
Block[{res, pf = 2},
res =
NestWhile[
{#[[1]] + step, #[[3]],
methodFunc[partialSumFunc, #[[1]] + step]} &,
{1, methodFunc[partialSumFunc, startStep],
methodFunc[partialSumFunc, startStep + step]},
Abs[N[#[[2]], pf*accGoal] - N[#[[3]], pf*accGoal]] >
10^-accGoal &];
(* n-steps, estimate, error *)
Append[res[[{1, 3}]],
Abs[N[res[[2]], pf*accGoal] - N[res[[3]], pf*accGoal]]]
];


Here is the application of the functions above:

sf[n_] := Sum[(-1)^i/i^3, {i, 1, n}]

(* partialSumFunc, accGoal, step, startStep, methodFunc *)
res = SearchSumValue[sf, 20, 30, 40, Shanks[#1, 6, #2] &];
N[res[[2]] - exact, 40]
(* 8.757707518816394163494370352471694535835*10^-28 *)

(* partialSumFunc, accGoal, step, startStep, methodFunc *)
res = SearchSumValue[sf, 50, 40, 40, Shanks[#1, 6, #2] &];
N[res[[2]] - exact, 50]
(* -3.1179069154714173193769908669678269443598103298019*10^-50 *)

• Richardson is really only as good as the auxiliary sequence (i.e. the abscissas used for constructing the underlying interpolating polynomial) you give it; if you give an appropriate auxiliary sequence, it can outdo most other convergence accelerators (see e.g. Romberg), and fail miserably otherwise. Thus, using Richardson takes a bit of finesse. Feb 16 '16 at 17:22
• Agreed, my general comments are in that line of thoughts. I mentioned the Richardson extrapolation mostly as a justification for the argument methodFunc in the search function SearchSumValue. Feb 16 '16 at 17:53
x = N[Sum[(-1)^n/n^3, {n, 1, Infinity}], 47]

y = NSum[(-1)^n/n^3, {n, 1, Infinity}, WorkingPrecision -> 47,
AccuracyGoal -> 47, PrecisionGoal -> 47, NSumTerms -> 10000]

z = NSum[(-1)^n/n^3, {n, 1, Infinity}, Method -> "AlternatingSigns",
WorkingPrecision -> 47, AccuracyGoal -> 47, PrecisionGoal -> 47]

x == y == z


True (-0.90154267736969571404980362113358749307373971926)

Using the simplest form:

e = NSum[(-1)^n/n^3, {n, 1, Infinity}]


we get with

e // FullForm


-0.9015426773696957

which is clearly less precise. But!

x === y === z === e


True

Strange ...

• Interesting, it seems my PrecisionGoal -> Infinty was causing some trouble. I see now that AccuracyGoal -> Infinity, PrecisionGoal -> 47 is better than AccuracyGoal -> 47, PrecisionGoal -> Infinty.
– user10957
Jun 7 '14 at 9:58
• @Jean-ClaudeArbaut - It is, of course, unsatisfactory, that you only find the "correct" number for NSumTerms (approx. 10000) by trial and error. Thanks for this interesting question.
– eldo
Jun 7 '14 at 10:01
• This doesn't directly answer the original question, but why not use the fact that Mathematica can evaluate the exact sum symbolically: Sum[(-1)^n/n^3, {n, 1, Infinity}] gives (-3*Zeta[3])/4. You may then use N, with a 2nd argument to specify the desired precision. Jun 7 '14 at 15:23
• @murray - for me thats THE answer if you can evaluate symbolically.
– eldo
Jun 7 '14 at 15:30
• @eldo: OK, I made my comment into an answer. Jun 7 '14 at 19:40

This doesn't directly answer the original question, but why not use the fact that Mathematica can evaluate the exact sum symbolically:

    s = Sum[(-1)^n/n^3, {n, 1, Infinity}]
(-3*Zeta[3])/4 *)


Then you may use N, with a 2nd argument to specify the desired precision, e.g.:

    N[s, 50]
(* -0.90154267736969571404980362113358749307373971925537 *)


(Of course such an approach will only work with those special series whose sums Mathematica already knows about somehow.)

• May I suggest that you split your answer into two lines: x = Sum[(-1)^n/n^3, {n, 1, Infinity}] // InputForm. Then: N[x, 100] (would make it clearer for the occasional reader).
– eldo
Jun 7 '14 at 20:02
• For a sum with alternating signs, be it numeric or symbolic, it is often worthwile to try summing the odd and even terms separately. In this case, both NSum calculate to the desired precision, and as a last step, just subtract. Jun 9 '14 at 20:51
• And there is a well-known ("Calculus II") Alternating Series Test whose proof implies that, for an alternating series whose terms have absolute values that decrease and approach 0, the (mathematical) truncation error in using the nth partial sum does not exceed the magnitude of the next term. Jun 10 '14 at 0:22
• @Wouter, but, as you know, this procedure is not always the best way to proceed (e.g. the classical alternating series for $\log 2$). Feb 14 '16 at 3:17

Edit: [I see that someone else gave you the same information I give in the following, about NSumTerms. I hope the following will at least reinforce the solution in your work.]

I found NSumTerms very helpful when Mathmatica becomes intransigent in my research, which is often! NSumTerms works well on your problem too. AlternatingSigns usually is necessary on stubborn alternating series that have no or very complicated closed forms like,

Sum[(1)^n (n^(1/n) - 1), {n, 1, Infinity}]


ie.

NSum[(-1)^x (x^(1/x) - 1), {x, 1, Infinity}, Method -> "AlternatingSigns", WorkingPrecision -> 80]


or

Sum[(-1)^x ((Log[x]/x)^n/n!), {x, 1, Infinity}, {n, 1, Infinity}]


ie.

 NSum[(-1)^x (Log[x]/x)^n/n!, {x, 1, Infinity}, {n, 1, Infinity},
Method -> "AlternatingSigns", WorkingPrecision -> 80,
PrecisionGoal -> 80, AccuracyGoal -> 80, NSumTerms -> 80]


. Both of the just mentioned codes give rise to approximations of the MRB constant, the most fruitful result of my research.

In your case, you need a whoping 7'th power of 10 terms, or there about, to get 80 accurate digits, like in the following:

NSum[(-1)^n/n^3, {n, 1, Infinity}, NSumTerms -> 10^7,
WorkingPrecision -> 80, AccuracyGoal -> 80,
PrecisionGoal -> Infinity]


gives

-0.9015426773696957140498036211335874930737397192553741613442036665063\ 7865433973482.

I am sure it is correct because if you compute the digits from the closed form, (as given above), like in the following...

N[-3/4 Zeta[3], 80]


you also get

-0.9015426773696957140498036211335874930737397192553741613442036665063\ 7865433973482

Alternating signs actually gives only 31 digits of accuracy in the following attempt.

 NSum[(-1)^n/n^3, {n, 1, Infinity}, NSumTerms -> 10^7,
Method -> "AlternatingSigns", WorkingPrecision -> 80,
AccuracyGoal -> 80, PrecisionGoal -> Infinity]


I know you want to use NSum but there are other options for accelerating the convergence of series.

Start by noting that your series

$\left\{-1,\frac{1}{8},-\frac{1}{27},\frac{1}{64},-\frac{1}{125},\frac{1}{216},-\frac{1}{343},...\right\}$

is the negative of this one

$\left\{1,-\frac{1}{8},\frac{1}{27},-\frac{1}{64},\frac{1}{125},-\frac{1}{216},\frac{1}{343},...\right\}$

which has the general term

$\frac{(-1)^k}{(k+1)^3}$

for k = 0, 1, 2,...

There is a little routine I came across which I thought was from the Borweins (see J.M.'s comment below), that I call the converger.

acc[n_]:=Module[{d,b,c,s},
d=(3+Sqrt[8])^n;
d=(d+1/d)/2;
b=-1;
c=-d;
s=0;
Table[c=b-c;s=s+c*a[k];b=(k+n)(k-n) b/((k+1/2)(k+1)),{k,0,n-1}];
s/d]


You prime it by entering the general term as a positive sequence:

a[k_]:=1/(k+1)^3


Now just run

-N[acc[300],200]


and you have about 200 places. The parameter n controls the number of iterations and hopefully the convergence. Quite fast and might get you out of jam when nothing else is working.

• That's the Cohen-Rodriguez Villegas-Zagier method for summing an alternating series. It's a very neat use of Chebyshev polynomials. Feb 16 '16 at 1:14

If you look at the FullForm or InputForm of the NSum result without using "AlternatingSigns" you will see that more digits are available; however, display is limited by the determined precision which is less than 17 digits. The determined precision which is implied by the shortened result also can be seen using Precision.

approx1 = NSum[(-1)^n/n^3, {n, 1, Infinity}, WorkingPrecision -> 24, AccuracyGoal -> 24, PrecisionGoal -> Infinity]


-0.9015426773696957

#[approx1] & /@ {Precision, FullForm, InputForm}


{16.774,-0.9015426773696956967432522482628641510116.773986512910163,-0.9015426773696956967432522482628641510116.773986512910163}

If the precision of the result is less than the requested precision then you should start investigating other methods.

approx2 = NSum[(-1)^n/n^3, {n, 1, Infinity}, WorkingPrecision -> 24, AccuracyGoal -> 24, PrecisionGoal -> Infinity, Method -> "AlternatingSigns"]


-0.901542677369693780504913

#[approx2] & /@ {Precision, FullForm, InputForm}


{24.1487,-0.90154267736969378050491347983360150530995076960472861836724.148671096657576,-0.901542677369693780504913479833601505309950769604728618367`24.148671096657576}

• Thanks. So that means you can get a result with lower precision than requested? (it's what I understand by playing around with this problem) I hoped there was a way to "just" get the requested precision, and Mma do all the stuff, but from links in comments above, I'm just starting to understand Mma numerics are much more complex than just "N" with number of digits. :-)
– user10957
Jun 7 '14 at 14:46