Question

Consider the following function $$f(x, n)=\sum_{i=1}^{n} \prod_{j=1}^{i} x^{n-j+1} .$$ a) Compute \(f(2,3)\) by hand. b) Write a Python code that computes \(f\). Verify \(f(2,3)\) matches your answer above.

Short answer

Question: Compute the value of \(f(2, 3)\) for the function $$f(x, n)=\sum_{i=1}^{n} \prod_{j=1}^{i} x^{n-j+1}.$$ Answer: When we manually calculate the function for \(f(2, 3)\), we get the value 104.

Step-by-step solution

Expand the summation and product

Let's expand the summation and product for \(f(2,3)\): $$f(2,3)=\sum_{i=1}^{3} \prod_{j=1}^{i} 2^{3-j+1}$$ Now, we break it down into individual terms for each value of \(i\).

Compute each term of the summation

When calculating each term, the product can be computed for different values of \(i\). 1. For \(i=1\), the product term is: \(2^{3-1+1}=2^3=8\) 2. For \(i=2\), the product term is: \((2^{3-2+1})(2^{3-1+1})=(2^2)(2^3)=4 \times 8=32\) 3. For \(i=3\), the product term is: \((2^{3-3+1})(2^{3-2+1})(2^{3-1+1})=(2^1)(2^2)(2^3)=2 \times 4 \times 8=64\)

Add the terms of the summation

Now, we sum all the terms we calculated in the previous step: $$f(2,3)=8+32+64=104$$ So, \(f(2,3)=104\). #b) Write a Python code that computes f. Verify f(2,3) matches your answer above# ```python def f(x, n): result = 0 for i in range(1, n+1): product_term = 1 for j in range(1, i+1): product_term *= x**(n-j+1) result += product_term return result f_2_3 = f(2, 3) print(f_2_3) ``` The Python code above computes the value of \(f(x,n)\). If you run this code, it will output `104`, which matches our manual calculation of \(f(2,3)\).

Key concepts

Summation and Product Notation

The mathematical concepts of summation and product notations are essential building blocks in numerical analysis. Summation, represented by the symbol \(\Sigma\), is used to denote the addition of a sequence of numbers. For instance, \(\sum_{i=1}^{n} a_i\) means that you should add together all values of \(a_i\) from \(i=1\) to \(i=n\).

Product notation, signified by the symbol \(\Pi\), denotes the multiplication of a sequence of factors. The expression \(\prod_{j=1}^{i} b_j\) means to multiply all values of \(b_j\) from \(j=1\) to \(j=i\). In the given exercise, \(f(x, n)\) uses both notations to define a complex function that sums the products of powers of \(x\). Breaking down these notations into their computational steps is crucial for understanding and solving the function by hand, as shown in the step-by-step solution.

Computing Mathematical Functions

Computing mathematical functions often involves dealing with sequences and series, and the ability to simplify expressions through summation and product notations. The function \(f(x, n)\) provided in the exercise requires us to perform a sequence of multiplications (products) before adding up the results (summations). Calculating \(f(2,3)\) by hand, we must carefully execute each step and pay special attention to the order of operations to ensure accuracy.

Firstly, expand the terms by handling the innermost operation: the product notation. After finding the product for each \(i\), we proceed by summing up these results to find the final value. This process is a practical example of how mathematical functions combine various operations, and thus it highlights the importance of a methodological approach to solving problems in numerical analysis.

Writing Python Code for Mathematical Computations

Translating mathematical functions into Python code can simplify and automate computations, which is particularly useful for complex or repetitive calculations. The function \(f(x, n)\) can be implemented in Python with the help of loops. The provided Python code demonstrates how to compute \(f(2, 3)\) using nested loops: an outer loop for the summation index \(i\) and an inner loop for the product index \(j\).

In the code snippet, the function \(f\) takes two arguments, \(x\) and \(n\). The outer loop accounts for the sum over \(i\), and for each iteration of this loop, an inner loop calculates the product of terms with increasing powers of \(x\). The code serves as an excellent educational tool, showcasing how mathematical notation can be systematically converted into a programming language to execute even the most intricate of functions faithfully.