Problem 12 - Advanced Engineering Mathematics

Question

Rewrite the given expression as a single power series whose general term involves \(x^{k}\). $$ \sum_{n=2}^{\infty} n(n-1) c_{n} x^{n}+2 \sum_{n=2}^{\infty} n(n-1) c_{n} x^{n-2}+3 \sum_{n=1}^{\infty} n c_{n} x^{n} $$

Step-by-Step Solution

Answer

The expression simplifies to a single power series: \( \sum_{n=0}^{\infty} f(n) x^{n} \) for some function \( f(n) \).

Step 1: Merge Like Series

To rewrite the given expression, notice the first two sums have identical terms except for a factor of 2 in the second term, and their powers of 2 differ by 2. We can merge these two by rewriting \[2\sum_{n=2}^{\infty} n(n-1) c_{n} x^{n-2} = \sum_{n=0}^{\infty} 2n(n+1) c_{n+2} x^{n}\].

Step 2: Adjust the Index for the Third Series

Start by changing the index of the sum \[ 3 \sum_{n=1}^{\infty} n c_{n} x^{n}\] to include all the terms, making it \[3 \sum_{n=0}^{\infty} (n+1) c_{n+1} x^{n}\].

Step 3: Combine All Series into a Single Series

Now, combine the modified series into one single series. The sum becomes:\[\sum_{n=0}^{\infty}\left(n(n-1) c_{n} + 2n(n+1) c_{n+2} + 3(n+1) c_{n+1}\right) x^{n}\].

Key Concepts

General Term

When dealing with power series, understanding the general term is crucial. A power series usually looks like this:

• \(\sum_{n=0}^{\infty} a_n x^n\)

This means it is a series where each term is made up of a coefficient, \(a_n\), and a power of \(x\), specifically \(x^n\). The general term describes a typical entry in this sequence.
Identifying and simplifying the general term ensures that the series can be adequately analyzed and worked with in a more flexible way.
In our exercise, once the series were combined, the general term became:

• \( \, n(n-1) c_{n} + 2n(n+1) c_{n+2} + 3(n+1) c_{n+1} \)

This allows us to see the relationship between terms in the series more clearly, facilitating further computations or evaluations.

Index Adjustment

The process of index adjustment involves changing the starting point or variable index of a series to simplify expressions or align multiple series. This technique can help unify series for easier manipulation.
In our particular exercise, we applied an index adjustment to streamline the third series. Initially starting from \(n=1\):

• \(3 \sum_{n=1}^{\infty} n c_{n} x^{n}\)

we shifted it to start from \(n=0\) for alignment with other series:

• \(3 \sum_{n=0}^{\infty} (n+1) c_{n+1} x^{n}\)

This change made it easier to merge this series with the others by presenting it with a unified format, starting points, and exponents of \(x\). Thus, index adjustment serves as a valuable tool for combining and simplifying series expressions.

Series Combination

Series combination is the process of merging several series into a single series. This can be advantageous because it allows us to treat multiple expressions under a single set of operations.
In the given exercise, the series combination involved:

• Merging two series with similar constructs but a factor of 2 difference in their second parts.
• Index adjustment for the third series to align it with others.

Finally, all series were represented as:

• \(\sum_{n=0}^{\infty}\left(n(n-1) c_{n} + 2n(n+1) c_{n+2} + 3(n+1) c_{n+1}\right) x^{n}\)

This procedure of combining series is essential for simplifying complex mathematical problems into a more straightforward format. This way, further calculation or evaluation can be focused on a single expression rather than multiple disconnected ones.