On the use of recurrent series in the search for roots of equations

Reading original papers can reveal  us “pearls” lost in time; moreover if they don’t  follow modern (and not so modern) standards of rigor. The master of this kind of results was Euler, who arrived at surprising results with amazing methods. The method we state here it's described by Euler in his 1748’s Introductio in Analysin Infinitorum.

 Let's mention two prior prerequisites:

[1] The first one is a well known recurrent series expansion (where the domain it holds…):

[2]The second one can also be easily deduced, we simply state it: 

 Now we have the necessary tools to explain the method. Suppose we have the following equation:

Suppose that all the roots are real and different. The above polynomium can be expressed in the following way:

Let be the following fraction:

The above fraction can be expressed as follows:

By [1] we know that the general term will be:

Let k be a very big number (infinitely big); because of the fact that the difference of numbers powers are bigger than the difference of that numbers, it will be a factor among

clearly bigger than the rest. Let p be the maximum of

Bearing in mind the statement before, we have 

and

 Therefore,

From that it follows that if we go on sufficiently through the serie 

the coefficient of each term divided by the preceding will give us the value p, and, consequently, the value of the smallest root, which will be equal to 1/p.

We can obtain with this method the value of the biggest root by means of the variable substitution x=1/z; in this case, the value p will give us directly the value of the biggest root.

We haven't mentioned the numerator coefficients because, indepently of their values, we will obtain the same value p. So, given

and bearing in mind [2], we can choose freely the values for 

EXAMPLE:

Suppose we want to obtain the biggest root of the following equation:

Let's deal with the variable change z=1/x and build the following fraction:

We choose arbritraily for

the values 1 and 2. The resulting series is: 1, 2, 7, 23, 76, 251, 829, 2738, ...

Therefore, the approximate value of the biggest root is 2738 / 829 ~ 3.3027744

Solving the equation in the traditional way we have as biggest root

which give as 3.3027756; value that exceeds our result by only a millionth.


For more info: Introductio in analysin infinitorum, chapter XIII