What causes this strange convergent sum?

Question

N[Sum[1/(x^2 + 1), {x, 1, Infinity}], 5]
N[Sum[1/(x^2 + x + 1), {x, 1, Infinity}], 5]

1.0767

0.79815 + 0.*10^-6 I

What causes the strange number?

Answer 1

Note that Mathematica can do the symbolic sums:

Sum[1/(x^2 + 1), {x, 1, Infinity}]

gives

$$ \frac{1}{2} (-1 + \pi \coth[\pi]) $$

and

Sum[1/(x^2 + x + 1), {x, 1, Infinity}]

gives

$$ -\frac{i \left(\psi ^{(0)}\left(1+\sqrt[3]{-1}\right)-\psi ^{(0)}\left(1-(-1)^{2/3}\right)\right)}{\sqrt{3}} $$ (where $\psi^{(0)}(x)$ is PolyGamma[0,x], and in fact if you apply FullSimplify this can be written in terms of HarmonicNumber as well). But you are asking for numeric versions of these to a specific number of significant figures. To that precision, the imaginary parts of the latter function do not quite cancel, and hence you get a small imaginary part.

Note also that using NSum[...] instead of N[Sum[...]] works better here, since it actually does the numerical sum by adding the terms, as opposed to calculating the symbolic form and then numerically evaluating it.

Answer 2

There are several methods in Sum which may help in various cases. To find your second sum symbolically in apparently real form you can try, e.g.

Sum[1/(1 + x + x^2), {x, 1, Infinity}, Method -> "HypergeometricTermPFQ"]

However there are still other ways to get the same result without methods specified, e.g. :

s = Sum[1/(1 + x + x^2), {x, 1, Infinity}];

FullSimplify[ ComplexExpand //@ s ]

or

FullSimplify @ ExpToTrig @ s

Note that we had to MapAll (shorthand //@) ComplexExpand i.e. apply the latter to every subexpression of s