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NSum

Status: Stable

documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

Description

NSum[f, {i, imin, imax}]
    gives a numerical approximation to the sum of f for i from imin to imax.

NSum[f, {i, imin, imax, di}] uses step di. imax may be Infinity. NSum[f, {i, ...}, {j, ...}, ...] evaluates a multidimensional sum (an inner bound may depend on an outer index). The index is localised (HoldAll). Method -> Automatic picks Euler-Maclaurin for monotone series, the Cohen-Villegas-Zagier method for alternating series, and Wynn's epsilon (partial-sum acceleration) otherwise; large finite sums use the difference of two infinite tails. Machine or arbitrary precision via WorkingPrecision.

Options: Method (Automatic | EulerMaclaurin | AlternatingSigns | WynnEpsilon), WorkingPrecision (default MachinePrecision), NSumTerms (head terms summed explicitly, default 15), NSumExtraTerms, WynnDegree, VerifyConvergence (default True; a divergent sum gives ComplexInfinity), AccuracyGoal, PrecisionGoal.

Examples

All examples below are verified against the current Mathilda build.

In[1]:= NSum[(-5)^i/i!, {i, 0, Infinity}, NSumTerms -> 25] - Exp[-5]
Out[1]= 1.43895e-15

In[2]:= NSum[1/i^2, {i, 1, Infinity}] - Pi^2/6 // N
Out[2]= 2.22045e-16

In[3]:= NSum[1/n^(11/10), {n, 1, Infinity}, WorkingPrecision -> 40] - Zeta[11/10]
Out[3]= 1.4693679385278593849609206715278070972733e-39

In[4]:= NSum[(-1)^x/(1 + (x - 12)^2), {x, 0, Infinity}, Method -> "AlternatingSigns", WorkingPrecision -> 30]
Out[4]= 0.2751938594139530395689715615907

In[5]:= NSum[1/2^i, {i, 0, Infinity, 2}]
Out[5]= 1.33333

In[6]:= NSum[Log[x]/x^(2 + 2 I), {x, 1, Infinity}]
Out[6]= -0.182175 - 0.136618*I

In[7]:= NSum[1/i^2, {i, 100, 10^6}]
Out[7]= 0.0100492

In[8]:= NSum[(-1)^n (2/n)^k/k^2, {n, 2, Infinity}, {k, 1, n}]
Out[8]= 0.770188

Implementation notes

Attributes: HoldAll, Protected.

Implementation status

Stable — documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

References

Notes & additional examples

Worked examples

In[1]:= NSum[1/n^2, {n, 1, Infinity}]
Out[1]= 1.64493

In[1]:= NSum[(-1)^(n+1)/n, {n, 1, Infinity}]
Out[1]= 0.693147

In[1]:= NSum[1/n^2, {n, 1, Infinity}, WorkingPrecision -> 30]
Out[1]= 1.644934066848226436472415166649

In[1]:= NSum[1/n^4, {n, 1, Infinity}, WorkingPrecision -> 30]
Out[1]= 1.082323233711138191516003696543

Notes

NSum[f, {i, imin, imax}] numerically sums a series, with imax allowed to be Infinity. The first two cases recover the Basel sum Pi^2/6 = 1.64493... and the alternating harmonic sum Log[2] = 0.693147.... With WorkingPrecision -> 30 the Basel sum is computed to 30 digits, and Sum[1/n^4] returns Pi^4/90 = 1.082323233711.... Method -> Automatic chooses Euler–Maclaurin for monotone series, the Cohen–Villegas–Zagier method for alternating series, and Wynn's epsilon otherwise. With VerifyConvergence -> True (default) a divergent sum gives ComplexInfinity.