(self, m=0, n=0, eps=0, eval_integral=True)
return s + iterm, abs(term)
Return an Euler-Maclaurin approximation of self, where m is the
        number of leading terms to sum directly and n is the number of
        terms in the tail.

        With m = n = 0, this is simply the corresponding integral
        plus a first-order endpoint correction.

        Returns (s, e) where s is the Euler-Maclaurin approximation
        and e is the estimated error (taken to be the magnitude of
        the first omitted term in the tail):

            >>> from sympy.abc import k, a, b
            >>> from sympy import Sum
            >>> Sum(1/k, (k, 2, 5)).doit().evalf()
            1.28333333333333
            >>> s, e = Sum(1/k, (k, 2, 5)).euler_maclaurin()
            >>> s
            -log(2) + 7/20 + log(5)
            >>> from sympy import sstr
            >>> print(sstr((s.evalf(), e.evalf()), full_prec=True))
            (1.26629073187415, 0.0175000000000000)

        The endpoints may be symbolic:

            >>> s, e = Sum(1/k, (k, a, b)).euler_maclaurin()
            >>> s
            -log(a) + log(b) + 1/(2*b) + 1/(2*a)
            >>> e
            Abs(1/(12*b**2) - 1/(12*a**2))

        If the function is a polynomial of degree at most 2n+1, the
        Euler-Maclaurin formula becomes exact (and e = 0 is returned):

            >>> Sum(k, (k, 2, b)).euler_maclaurin()
            (b**2/2 + b/2 - 1, 0)
            >>> Sum(k, (k, 2, b)).doit()
            b**2/2 + b/2 - 1

        With a nonzero eps specified, the summation is ended
        as soon as the remainder term is less than the epsilon.
Return an Euler-Maclaurin approximation of self, where m is the
        number of leading terms to sum directly and n is the number of
        terms in the tail.
Return an Euler - Maclaurin approximation of self where m is the number of leading terms to sum directly and n is the number of terms in the tail .
def euler_maclaurin(self, m=0, n=0, eps=0, eval_integral=True):
        """
        Return an Euler-Maclaurin approximation of self, where m is the
        number of leading terms to sum directly and n is the number of
        terms in the tail.

        With m = n = 0, this is simply the corresponding integral
        plus a first-order endpoint correction.

        Returns (s, e) where s is the Euler-Maclaurin approximation
        and e is the estimated error (taken to be the magnitude of
        the first omitted term in the tail):

            >>> from sympy.abc import k, a, b
            >>> from sympy import Sum
            >>> Sum(1/k, (k, 2, 5)).doit().evalf()
            1.28333333333333
            >>> s, e = Sum(1/k, (k, 2, 5)).euler_maclaurin()
            >>> s
            -log(2) + 7/20 + log(5)
            >>> from sympy import sstr
            >>> print(sstr((s.evalf(), e.evalf()), full_prec=True))
            (1.26629073187415, 0.0175000000000000)

        The endpoints may be symbolic:

            >>> s, e = Sum(1/k, (k, a, b)).euler_maclaurin()
            >>> s
            -log(a) + log(b) + 1/(2*b) + 1/(2*a)
            >>> e
            Abs(1/(12*b**2) - 1/(12*a**2))

        If the function is a polynomial of degree at most 2n+1, the
        Euler-Maclaurin formula becomes exact (and e = 0 is returned):

            >>> Sum(k, (k, 2, b)).euler_maclaurin()
            (b**2/2 + b/2 - 1, 0)
            >>> Sum(k, (k, 2, b)).doit()
            b**2/2 + b/2 - 1

        With a nonzero eps specified, the summation is ended
        as soon as the remainder term is less than the epsilon.
        """
        m = int(m)
        n = int(n)
        f = self.function
        if len(self.limits) != 1:
            raise ValueError("More than 1 limit")
        i, a, b = self.limits[0]
        if (a > b) == True:
            if a - b == 1:
                return S.Zero,S.Zero
            a, b = b + 1, a - 1
            f = -f
        s = S.Zero
        if m:
            if b.is_Integer and a.is_Integer:
                m = min(m, b - a + 1)
            if not eps:
                for k in range(m):
                    s += f.subs(i, a + k)
            else:
                term = f.subs(i, a)
                if term:
                    test = abs(term.evalf(3)) < eps
                    if test == True:
                        return s, abs(term)
                    elif not (test == False):
                        # a symbolic Relational class, can't go further
                        return term, S.Zero
                s += term
                for k in range(1, m):
                    term = f.subs(i, a + k)
                    if abs(term.evalf(3)) < eps:
                        return s, abs(term)
                    s += term
            if b - a + 1 == m:
                return s, S.Zero
            a += m
        x = Dummy('x')
        I = C.Integral(f.subs(i, x), (x, a, b))
        if eval_integral:
            I = I.doit()
        s += I

        def fpoint(expr):
            if b is S.Infinity:
                return expr.subs(i, a), 0
            return expr.subs(i, a), expr.subs(i, b)
        fa, fb = fpoint(f)
        iterm = (fa + fb)/2
        g = f.diff(i)
        for k in xrange(1, n + 2):
            ga, gb = fpoint(g)
            term = C.bernoulli(2*k)/C.factorial(2*k)*(gb - ga)
            if (eps and term and abs(term.evalf(3)) < eps) or (k > n):
                break
            s += term
            g = g.diff(i, 2, simplify=False)
        return s + iterm, abs(term)
def euler_maclaurin ( self , m = 0 , n = 0 , eps = 0 , eval_integral = True ) : m = int ( m ) n = int ( n ) f = self . function if len ( self . limits ) != 1 : raise ValueError ( "More than 1 limit" ) i , a , b = self . limits [ 0 ] if ( a > b ) == True : if a - b == 1 : return S . Zero , S . Zero a , b = b + 1 , a - 1 f = - f s = S . Zero if m : if b . is_Integer and a . is_Integer : m = min ( m , b - a + 1 ) if not eps : for k in range ( m ) : s += f . subs ( i , a + k ) else : term = f . subs ( i , a ) if term : test = abs ( term . evalf ( 3 ) ) < eps if test == True : return s , abs ( term ) elif not ( test == False ) : # a symbolic Relational class, can't go further return term , S . Zero s += term for k in range ( 1 , m ) : term = f . subs ( i , a + k ) if abs ( term . evalf ( 3 ) ) < eps : return s , abs ( term ) s += term if b - a + 1 == m : return s , S . Zero a += m x = Dummy ( 'x' ) I = C . Integral ( f . subs ( i , x ) , ( x , a , b ) ) if eval_integral : I = I . doit ( ) s += I def fpoint ( expr ) : if b is S . Infinity : return expr . subs ( i , a ) , 0 return expr . subs ( i , a ) , expr . subs ( i , b ) fa , fb = fpoint ( f ) iterm = ( fa + fb ) / 2 g = f . diff ( i ) for k in xrange ( 1 , n + 2 ) : ga , gb = fpoint ( g ) term = C . bernoulli ( 2 * k ) / C . factorial ( 2 * k ) * ( gb - ga ) if ( eps and term and abs ( term . evalf ( 3 ) ) < eps ) or ( k > n ) : break s += term g = g . diff ( i , 2 , simplify = False ) return s + iterm , abs ( term )