Problem 43 - Thomas Calculus

Question

Find the sum of each series in Exercises \(41-48\) $$ \sum_{n=1}^{\infty} \frac{40 n}{(2 n-1)^{2}(2 n+1)^{2}} $$

Step-by-Step Solution

Answer

Use partial fraction decomposition to find simplified terms and then sum them for the series solution.

Step 1: Write the Series Formula

The given series is \( \sum_{n=1}^{\infty} \frac{40n}{(2n-1)^2(2n+1)^2} \). Our task is to find the sum of this series. Each term in the series has the form \( \frac{40n}{(2n-1)^2(2n+1)^2} \).

Step 2: Analyze the General Term

Consider the term \( \frac{40n}{(2n-1)^2(2n+1)^2} \). Notice that both factors in the denominator can be expressed in a manner that simplifies using partial fraction decomposition. These are the squared terms \((2n-1)^2\) and \((2n+1)^2\).

Step 3: Perform Partial Fraction Decomposition

Express the general term as a sum of simpler fractions. This can be done by setting \[ \frac{40n}{(2n-1)^2(2n+1)^2} = \frac{A}{2n-1} + \frac{B}{(2n-1)^2} + \frac{C}{2n+1} + \frac{D}{(2n+1)^2} \] and solving for the coefficients \(A, B, C,\) and \(D\).

Step 4: Solve for the Coefficients

Set up and solve the system of equations derived from the numerator equality to find suitable coefficients \(A, B, C,\) and \(D\). You'll end up solving usually by substituting values or by comparing coefficients.

Step 5: Sum the Decomposed Series

Once the partial fraction decomposition is complete and coefficients determined, each part of the decomposition will be a known series type. Often, such series have known sums. Compute these sums individually.

Step 6: Verify and Calculate Final Sum

Add up the sums of the decomposed series parts to find the sum of the entire series. Make sure to include all parts as derived from partial fraction decomposition.

Step 7: Solution Verification

Review each calculation and verify through substitution or checks with simpler cases if applicable. Confirm that the derived expressions and final summed result align with expected known results or behaviors of convergent geometric or arithmetic series.

Key Concepts

Partial Fraction Decomposition

Partial fraction decomposition is a technique used in algebra to express a complex rational expression as a sum of simpler fractions. This approach makes it easier to work with the expression, especially when integrating or summing a series. In our given series, the term is \( \frac{40n}{(2n-1)^2(2n+1)^2}\).
These terms, \( (2n-1)^2 \) and \( (2n+1)^2 \), are squared, making the series initially hard to tackle.

By decomposing the fraction, we transform the expression into a form that is more manageable. We essentially want to break it down into:

• A term over \( 2n-1 \)
• A term over \( (2n-1)^2 \)
• A term over \( 2n+1 \)
• A term over \( (2n+1)^2 \)

To find the coefficients \(A, B, C,\) and \(D\), set up equations by matching terms on both sides with the complex fraction. Substitute values or compare coefficients accordingly.
This breakdown simplifies each term in the series, making subsequent computation of the sum more straightforward.

Convergent Series

A convergent series is an infinite series that approaches a certain value as more terms are added. Not all series are convergent; some diverge, meaning their sums continue to grow without bounds as more terms are included.

In this context, we are dealing with the series \[ \sum_{n=1}^{\infty} \frac{40n}{(2n-1)^2(2n+1)^2} \]. A crucial step in handling such series is checking for convergence. Without this, our effort to find an actual sum might be in vain.

• If a series converges, it has a finite sum.
• If it diverges, this sum does not exist.

For a series involving partial fractions like this, each decomposed part typically can be checked for convergence separately.
The convergence guarantees that combining these parts will yield a meaningful result, reflecting an actual sum for the series in question. Techniques like the ratio test, comparison test, or using known convergent series for reference often help in these cases.

Series Summation Technique

Once the partial fraction decomposition is done and the convergence of the series is established, we need a technique to compute the sum of the series. Using known sums and properties of specific simpler series is often key.
Because the decomposition allows us to rewrite the original term as simpler fractions, each corresponds to a series type that is more familiar.

For instance:

• Geometric series have explicit formulas for their sums.
• Telescope series have terms cancel each other out, simplifying the summation.

Each part of the decomposition is then summed as an individual series. You add up these resulting sums to obtain the total sum of the original series.
Applying the correct series summation technique requires identifying the type of each "sub-series" from the partial fractions. Use this identification to leverage known sums and properties.
Verifying the sum ensures accuracy, confirming the manual calculations align with expected results, and adhere to mathematical expectations for convergent series.