Problem 10 - Advanced Engineering Mathematics

Question

Rewrite the given power series so that its general term involves \(x^{k}\). $$ \sum_{n=3}^{\infty}(2 n-1) c_{n} x^{n-3} $$

Step-by-Step Solution

Answer

\(\sum_{k=0}^{\infty} (2k + 5) c_{k+3} x^k\).

Step 1: Identify the General Term

The given power series is \( \sum_{n=3}^{\infty} (2n-1) c_n x^{n-3} \). Let's identify the general term, which is \((2n-1) c_n x^{n-3}\). We are asked to rewrite it in terms of \(x^k\).

Step 2: Re-index the Series

To rewrite in terms of \(x^k\), we set \(k = n - 3\). This implies that \(n = k + 3\). As \(n\) starts at 3, substituting this into the expression gives \(k = 0\).

Step 3: Substitute and Simplify

Replace \(n\) with \(k+3\) in the general term: \((2(k+3)-1) c_{k+3} x^{k}\). Simplifying this expression gives \((2k + 6 - 1) c_{k+3} x^k = (2k + 5)c_{k+3}x^k\).

Step 4: Rewrite the Series

Substitute the re-indexed term back into the series, starting from \(k = 0\):\[\sum_{k=0}^{\infty} (2k + 5) c_{k+3} x^k\] This is the power series rewritten in terms of \(x^k\).

Key Concepts

Re-indexing Series

Re-indexing a series is a crucial technique used when you want to change the variables in a series to make calculations or interpretations easier.
In the given exercise, the series is initially in the form:

• \( \sum_{n=3}^{\infty} (2n-1) c_n x^{n-3} \)

To re-index this series, we introduce a new variable, \(k\), which we set equal to \(n - 3\). This effectively shifts the index of summation and allows us to express the series in terms of \(x^k\).
In other words:

• Let \(k = n-3\), giving: \(n = k + 3\)
• If \(n\) starts from 3, then \(k\) must start from 0

This process not only shifts the series' index base but also makes it more flexible to work with, especially when solving more complex problems involving power series.

General Term Identification

Identifying the general term of a series is like finding the building blocks of a series. Each series has a specific pattern that repeats for every term, and your goal is to find this pattern.
The original series in our case:

• \((2n-1) c_n x^{n-3}\)

The general term here includes coefficients that are determined by the expressions provided.
By recognizing these coefficients, we know what manipulates each term's power of \(x\) and what multiplies the part involving \(c_n\).
This makes it possible to re-structure or transform the series effectively.

• For the rewritten series, the new general term is: \((2k + 5) c_{k+3} x^k\)
• This is obtained by substituting \(n\) with \(k + 3\) and simplifying the expression

Understanding and correctly altering this general term is key in transforming series termed with \(x^n\) to those in terms of \(x^k\). It reflects the character and progression of the given series.

Mathematical Series Simplification

Simplifying a mathematical series primarily involves reducing complexity to make further calculations more straightforward.
After re-indexing and identifying the general term, the next step is to substitute and simplify the expressions within to obtain a cleaner, more understandable form of the series.

• Insert \(n = k + 3\) into the identified general term \((2n-1) c_n x^{n-3}\)
• This substitution yields: \((2(k+3)-1) c_{k+3} x^{k}\)
• Upon simplifying, it results to \((2k+5)c_{k+3}x^k\)

The last piece of the puzzle is to recreate the series using the re-indexed terms:

• Thus, the series now reads: \(\sum_{k=0}^{\infty} (2k + 5) c_{k+3} x^k\)

This form is not only easier to handle but often required in problems or proofs involving power series where certain properties or results can be more readily applied.