Problem 75 - Calculus Early Transcendentals

Question

Let $$f(x)=\sum_{k=0}^{\infty} c_{k} x^{k} \quad \text { and } \quad g(x)=\sum_{k=0}^{\infty} d_{k} x^{k}$$ a. Multiply the power series together as if they were polynomials, collecting all terms that are multiples of \(1, x,\) and \(x^{2} .\) Write the first three terms of the product \(f(x) g(x)\) b. Find a general expression for the coefficient of \(x^{n}\) in the product series, for \(n=0,1,2, \ldots\)

Step-by-Step Solution

Answer

Based on the given step by step solution, find a general expression for the coefficient of \(x^n\) in the product series of the two power series, \(f(x) = \sum_{k=0}^{\infty} c_k x^k\) and \(g(x) = \sum_{k=0}^{\infty} d_k x^k\). The general expression for the coefficient of \(x^n\) in the product series is \(\sum_{k=0}^{n} c_k d_{n-k}\).

Step 1: Define the power series

Given the power series, we have: $$ f(x) = \sum_{k=0}^{\infty} c_k x^k \\ g(x) = \sum_{k=0}^{\infty} d_k x^k $$

Step 2: Write the product of the two power series

The product of these power series will be: $$ f(x)g(x) = \left(\sum_{k=0}^{\infty} c_k x^k \right)\left(\sum_{m=0}^{\infty} d_m x^m \right) $$

Step 3: Expand the product of the power series and collect terms

To collect the multiples of \(1, x,\) and \(x^2\), we need to expand the product: $$ f(x)g(x) = \left(c_0x^0 + c_1x^1 + c_2x^2 + \cdots \right) \left(d_0x^0 + d_1x^1+ d_2x^2 + \cdots \right) $$ When expanded, we get: 1. Coefficient of \(x^0\): \(c_0d_0\) 2. Coefficient of \(x^1\): \(c_0d_1 + c_1d_0\) 3. Coefficient of \(x^2\): \(c_0d_2 + c_1d_1 + c_2d_0\) So, the first three terms of the product are: $$ f(x)g(x) = (c_0d_0) + (c_0d_1 + c_1d_0) x + (c_0d_2 + c_1d_1 + c_2d_0) x^2 + \cdots $$

Step 4: Find the general expression for the coefficient of \(x^n\) in the product series

For a general expression for the coefficient of \(x^n\), we note that we need to sum over all possible combinations of \(k\) and \(m\) that result in the exponent \(n\). Therefore, we have: $$ \text{Coefficient of } x^n = \sum_{k=0}^{n} c_k d_{n-k} $$ This general expression can be used to find the coefficient of \(x^n\) in the product series for any given value of \(n\).

Key Concepts

Coefficient Calculation

In the realm of power series, calculating the coefficients is a crucial part of understanding the series' behavior. A power series is expressed as \( f(x) = \sum_{k=0}^{\infty} c_k x^k \), where \( c_k \) are coefficients of the series. When dealing with the product of two power series, \( f(x) \) and \( g(x) \), the task is to find the coefficients of the resulting series from their multiplication.To find the coefficient of \( x^n \) in the product \( f(x)g(x) \), note that it comes from summing terms where the indices sum to \( n \). The general rule for the coefficient of \( x^n \) is:

• Sum over all pairs of indices \( (k, m) \) such that \( k + m = n \).
• For each pair, multiply \( c_k \) from \( f(x) \) by \( d_m \) from \( g(x) \).
• The expression becomes \( \sum_{k=0}^{n} c_k d_{n-k} \).

This formula is a cornerstone for understanding more complex series manipulations.

Series Multiplication

Multiplying power series is akin to multiplying polynomials, though it goes to infinity. The power series are given by:

• \( f(x) = \sum_{k=0}^{\infty} c_k x^k \)
• \( g(x) = \sum_{k=0}^{\infty} d_k x^k \)

To find the product \( f(x)g(x) \), consider expanding these series similar to how one distributes terms in polynomial multiplication:- Multiply each term of \( f(x) \) by each term of \( g(x) \).- Collect like terms that have the same power of \( x \).For example, to find the first three terms, you gather:

• Term for \( x^0 \): Comes solely from \( c_0d_0 \).
• Term for \( x^1 \): From \( c_0d_1 \) and \( c_1d_0 \). Thus, \( c_0d_1 + c_1d_0 \).
• Term for \( x^2 \): From \( c_0d_2 \), \( c_1d_1 \), and \( c_2d_0 \). Resulting in \( c_0d_2 + c_1d_1 + c_2d_0 \).

This systematic approach leads you to the general expression for any term of the product series.

Calculus Step-by-Step

Working through calculus problems step by step ensures precision and deep understanding. When operating on power series, similar to the one in our exercise:1. **Definition**: Start by clearly writing down the power series expressions for \( f(x) \) and \( g(x) \).
2. **Setup**: Arrange the multiplication \( f(x)g(x) \) so each series term interacts with each other, akin to matrix multiplication but applied to series.
3. **Expansion**: Expand these products by multiplying terms like polynomials. Keep in mind that every pair \((c_k, d_m)\) translates to an \(x^n\) term when \(k + m = n\).
4. **Collection**: Gather and sum like terms to simplify and establish the coefficient for each power of \(x\). The process can be repetitive but ensures each step builds on the last, forming a clear path to the solution.By following these meticulous steps, even the most complex series multiplication becomes manageable, laying a solid foundation for further explorations in calculus.