Question

Use the difference method to sum the series $$ \sum_{n=2}^{N} \frac{2 n-1}{2 n^{2}(n-1)^{2}} $$

Short answer

Use partial fractions, decompose the sum and apply the difference method.

Step-by-step solution

Decompose the General Term

Start by breaking down the general term. Given term is: \( \frac{2n-1}{2n^2(n-1)^2} \). Perform partial fraction decomposition to express it in simpler terms.

Apply Partial Fraction Decomposition

Find constants A, B, C, and D such that: \[ \frac{2n-1}{2n^2(n-1)^2} = \frac{A}{n-1} + \frac{B}{(n-1)^2} + \frac{C}{n} + \frac{D}{n^2} \]. Solve for A, B, C, and D by equating coefficients.

Substitute Constants Back into the Series

Substitute the constants found in Step 2 back into the original sum: \[ \sum_{n=2}^{N} \left( \frac{A}{n-1} + \frac{B}{(n-1)^2} + \frac{C}{n} + \frac{D}{n^2} \right) \].

Simplify the Series

Split the series into four separate sums and simplify each part: \[ A \sum_{n=2}^{N} \frac{1}{n-1} + B \sum_{n=2}^{N} \frac{1}{(n-1)^2} + C \sum_{n=2}^{N} \frac{1}{n} + D \sum_{n=2}^{N} \frac{1}{n^2} \]. Use the properties of sums for logarithmic and polynomial series.

Evaluate Each Sum

Evaluate each sum separately: \( A \sum_{n=2}^{N} \frac{1}{n-1}, B \sum_{n=2}^{N} \frac{1}{(n-1)^2}, C \sum_{n=2}^{N} \frac{1}{n}, \text{and} D \sum_{n=2}^{N} \frac{1}{n^2} \).

Apply the Difference Method

Recognize that \( \sum_{n=2}^{N} \frac{1}{n-1} \) and \( \sum_{n=2}^{N} \frac{1}{n} \) are telescoping series. Similarly simplify \( \sum_{n=2}^{N} \frac{1}{(n-1)^2} \) and \( \sum_{n=2}^{N} \frac{1}{n^2} \).

Combine Results

Sum the results obtained from each series to get the overall sum of the series. Combine all the simplified terms.

Final Result

Substitute values of N and perform the final calculation to find the closed form of the given series.

Key concepts

partial fraction decomposition

Partial fraction decomposition is a technique used to break down complex rational expressions into simpler fractions that are easier to handle. It is particularly useful when dealing with polynomial fractions. This method works by expressing the complicated fraction as a sum of simpler fractions.

For example, consider a fraction like \(\frac{P(x)}{Q(x)}\), where \(P(x)\) and \(Q(x)\) are polynomials. Decomposing it involves finding constants A, B, C, etc. such that:
\[ \frac{P(x)}{Q(x)} = \frac{A}{x-r_1} + \frac{B}{(x-r_2)} + \ldots + \frac{C}{(x-r_n)}\]

In the given problem, we decompose \(\frac{2n-1}{2n^2(n-1)^2}\) into parts using constants A, B, C, and D. Finding these constants involves setting up an equation and solving for the coefficients by comparing the terms on both sides. This step can be tricky but is crucial for making the expression simpler to sum.

telescope series

A telescoping series is a series where most of the terms cancel out when the series is summed, leaving behind a much simpler expression. This property makes it easier to find the sum of long or otherwise complicated series.

For instance, in the series \(\sum_{n=2}^{N} \frac{1}{n-1}\), if you write out a few terms, you'll notice they cancel out:
\[ \frac{1}{1} + \frac{1}{2} + \frac{1}{3} - \cdots - \frac{1}{N} \]
This series 'telescopes' down to just a few remaining terms.

In our exercise, recognizing the telescoping nature in terms like \(\sum_{n=2}^{N} \frac{1}{n-1}\) makes the series much easier to sum. Most terms cancel out, leaving us with simple expressions to evaluate.

polynomial series

Polynomials appear frequently in series and sums. A polynomial series can often be simplified by recognizing patterns like the sum of squares or the sum of cubes.

For example, the sum \(\sum_{n=1}^N n^2\) can be represented using the formula for the sum of squares. Similarly:
\[ \sum_{n=1}^N n^2 = \frac{N(N+1)(2N+1)}{6} \]
These formulas help us simplify and compute sums involving polynomial terms quickly.

The given problem involves polynomial terms within a fractional series. Decomposing these into partial fractions and recognizing them as potential polynomial sums helps break down complex sums into manageable components.

logarithmic series

A logarithmic series is a series representation of logarithms or involves terms that can be related to logarithms. For instance, the harmonic series \(H_N = \sum_{n=1}^N \frac{1}{n}\) is closely related to the natural logarithm.

As \(N\) grows large, the harmonic series approximates the natural logarithm:
\[ H_N \approx \ln(N) + \gamma \]
where \(\gamma\) is the Euler-Mascheroni constant.

In the context of our problem, once we simplify the fractions and recognize some series as harmonic, we can use properties of logarithms to simplify the sum. This is evident when you sum terms like \(\sum_{n=2}^{N} \frac{1}{n-1}\) and \(\sum_{n=2}^{N} \frac{1}{n}\).