# Sage incorrectly evaluates series

It incorrectly evaluates $\displaystyle\sum_{n=0}^{\infty}\frac{1}{((2n+1)^2-4)^2}=\frac{\pi^2}{64}-\frac{1}{12}$, but correct answer is $\displaystyle\frac{\pi^2}{64}$

Sage incorrectly evaluates series

2

Indeed:

```
sage: n = var('n')
sage: sum(1/((2*n+1)^2-4)^2, n, 0, Infinity)
1/64*pi^2 - 1/12
```

See ticket #22005. Mathematica does it correctly:

```
sage: sum(1/((2*n+1)^2-4)^2, n, 0, Infinity, algorithm='mathematica')
1/64*pi^2
```

Giac gives this:

```
sage: sum(1/((2*n+1)^2-4)^2, n, 0, Infinity, algorithm='giac')
1/32*Psi(-1/2, 1) - 1/8
```

And SymPy seems to do it correctly:

```
sage: from sympy.abc import n
sage: from sympy import summation, oo
sage: A = summation(1/((2*n+1)^2-4)^2, (n, 0, oo))
sage: A._sage_()
1/64*pi^2
```

I created ticket #22004 so that one can do:

```
sage: sum(1/((2*n+1)^2-4)^2, n, 0, Infinity, algorithm='sympy')
1/64*pi^2
```

This may actually be https://sourceforge.net/p/maxima/bugs... which is apparently fixed in upstream.

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Asked: ** 2016-12-01 09:50:57 +0200 **

Seen: **162 times**

Last updated: **Dec 01 '16**

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Thanks for reporting !