How to resolve the algorithm Formal power series step by step in the Mathematica/Wolfram Language programming language

Problem Statement
Step by Step Solution
Sourcecode

Problem Statement

A power series is an infinite sum of the form

⋅ x +

⋅

⋯

{\displaystyle a_{0}+a_{1}\cdot x+a_{2}\cdot x^{2}+a_{3}\cdot x^{3}+\cdots }

The ai are called the coefficients of the series. Such sums can be added, multiplied etc., where the new coefficients of the powers of x are calculated according to the usual rules. If one is not interested in evaluating such a series for particular values of x, or in other words, if convergence doesn't play a role, then such a collection of coefficients is called formal power series. It can be treated like a new kind of number. Task: Implement formal power series as a numeric type. Operations should at least include addition, multiplication, division and additionally non-numeric operations like differentiation and integration (with an integration constant of zero). Take care that your implementation deals with the potentially infinite number of coefficients. As an example, define the power series of sine and cosine in terms of each other using integration, as in

sin ⁡ x

∫

cos ⁡ t

d t

{\displaystyle \sin x=\int _{0}^{x}\cos t,dt}

cos ⁡ x

1 −

∫

sin ⁡ t

d t

{\displaystyle \cos x=1-\int _{0}^{x}\sin t,dt}

Goals: Demonstrate how the language handles new numeric types and delayed (or lazy) evaluation.

Let's start with the solution:

Step by Step solution about How to resolve the algorithm Formal power series step by step in the Mathematica/Wolfram Language programming language

The provided Wolfram code calculates the series expansions for the cosine and sine functions, subtracts the sine from the integral of the cosine, and evaluates the result. Here's a detailed explanation of each part:

Series[Cos[x], {x, 0, 10}]: This line calculates the series expansion of the cosine function around x = 0 up to the 10th order. The syntax Series[f[x], {x, a, n}] generates a series expansion of the function f[x] around the point x = a up to the nth order. In this case, it computes the Taylor series representation of the cosine function with coefficients up to the 10th power of x.
Series[Sin[x], {x, 0, 8}]: This line calculates the series expansion of the sine function around x = 0 up to the 8th order. It follows the same syntax and generates a Taylor series representation of the sine function with coefficients up to the 8th power of x.
sin - Integrate[cos, x]:** This line subtracts the sine function from the integral of the cosine function. The Integrate[f[x], x] operation computes the integral of the function f[x] with respect to x. Here, the integral of the cosine function is calculated, which is the sine function up to a constant of integration. By subtracting the sine from this integral, the constant of integration is eliminated.

The result of this evaluation is the expression for the difference between the sine and the integral of the cosine. It provides a polynomial approximation of this difference function in terms of powers of x.

Source code in the wolfram programming language

cos = Series[Cos[x], {x, 0, 10}];
sin = Series[Sin[x], {x, 0, 8}];
sin - Integrate[cos, x]