Calculating Products in Mathematics

Functions for Calculating Products and Sums

• Product Command: Used to determine a formula for the product of terms.
• Mul Command: Calculates the finite product, similar to the add command for summation.
• Usage Recommendation: Prefer mul for calculating explicit products of terms.

Examples of Summation and Product Commands

• Summation Example: The sum of the first 10 integers is calculated with add(i,i=1..10); yielding 55.
• Formula for Summation: The sum formula for the first n integers is sum(i,i=1..n); which results in ((n+1)^2/2 - n^2/2).
• Product of Integers: The product of the first 10 integers is mul(i,i=1..10); resulting in 3,628,800.
• Product Formula: The product formula for the first n integers is given by product(i,i=1..n); which equals (\Gamma(n + 1)); Gamma function outlined under ?GAMMA.

Command Structure

• First Argument: Represents terms to sum or multiply.
• Second Argument: Indicates the index and the range over which the operation is performed.

Handling Sequences

• Example with Lists: To add a sequence from a list L, use add(L[i],i=1..4); resulting in 16.

Notes on Maple Calculation

• If a formula cannot be computed, Maple will return the function call.
• Example of Failure: For sum(n^k/(n-1),n=2..k);, Maple cannot produce a closed-form formula.

Infinite Sums and Products

• Infinite Sum Example: sum(1/x^2,x=1..infinity); equals (\pi^2/6).
• Infinite Product Example: product(x^n,x=1..infinity); computes product over an infinite range.

Inert Form of the Sum Command

• Inert Sum: Use Sum to represent sums without calculating them immediately; for example, S := Sum(x^k,k=0..n);.
• Expression Example: X := 2 + 3S - S^3; demonstrates an expression involving an inert sum.
• Value Command: To evaluate an inert sum, use value(S);.