public static Rational Harmonic(BigInteger n)
{
Rational sum = 0;
for (BigInteger k = 1; k <= n; k++)
{
sum += Rational.Inv(k);
}
return(sum);
}
public static Rational Harmonic(BigInteger n, int m)
{
Rational sum = 0;
for (BigInteger k = 1; k <= n; k++)
{
sum += Rational.Inv(Rational.Pow(k, m));
}
return(sum);
}
public Series Inv()
{
// We have the following recursive formula for the coefficients b[n] of
// the multiplicative inverse f(z)^-1:
//
// b[0] = a[0]^-1
// b[1] = -a[0]^-1(a[1]b[0])
// b[2] = -a[0]^-1(a[1]b[1]+a[2]b[0])
// b[3] = -a[0]^-1(a[1]b[2]+a[2]b[1]+a[3]b[0])
// ...
//
// We also have an explicit formula for b[n]:
//
// b[n] = a[0]^-1(1-c[n,1]+c[n,2]-c[n,3]+...+c[n,n]),
// c[n,k] = a[0]^-k sum(multinomial*a[i[1]]...a[i[k]])
// where the sum is taken over all partitions (i[1],...,i[k]) of n
// into k parts, and multinomial is the multinomial coefficient for the
// partition.
return(new Series((n, s) =>
{
Rational a0inv = Rational.Inv(this[0]);
if (n == 0)
{
return a0inv;
}
// Always prefer the recursive formula because it is much faster.
if (s.ComputeTo(n - 1))
{
int high = n;
Rational c = Rational.Zero;
if (high > this.Degree)
{
high = this.Degree;
}
for (int i = 1; i <= high; i++)
{
c += this[i] * s[n - i];
}
return -a0inv * c;
}
else
{
Rational m = a0inv;
Rational c = Rational.Zero;
for (int i = 1; i <= n; i++)
{
m *= -a0inv;
c += m * PowCoefficient(this, i, n, true);
}
return c;
}
}, order: 0, degree: this.Degree != 0 ? InfinityDegree : 0, cache: true));
}