Proving the Convergence of a Series Using Partial Fractions

In this article, we will demonstrate the use of partial fractions to prove the convergence of a specific infinite series. Specifically, we aim to show that the sum of the series (sum_{n1}^{infty} frac{1}{nn1n2}) converges to (frac{1}{4}). This process involves several steps, including partial fraction decomposition, solving linear equations, and evaluating the resulting series.

Step 1: Partial Fraction Decomposition

The first step is to express (frac{1}{nn1n2}) as a sum of simpler fractions. We assume that:

[frac{1}{nn1n2} frac{A}{n} frac{B}{n1} frac{C}{n2}]

Multiplying both sides by the denominator (nn1n2) gives the equation:

[1 An1n2 Bnn2 Cnn1]

Step 2: Expanding and Finding Coefficients

To find the constants (A), (B), and (C), we expand and compare coefficients:

[1 A(n^2 - 3n 2) B(n^2 - 2n) C(n^2 - n)]

Combining like terms, we get:

[1 A n^2 - 3A n 2A B n^2 - 2B n C n^2 - C n]

Grouping the terms, we have:

[1 (A B C)n^2 (-3A - 2B - C)n (2A)]

Setting the coefficients on both sides of the equation equal to each other, we obtain the following system of linear equations:

[begin{cases} A B C 0 -3A - 2B - C 0 2A 1 end{cases}]

Step 3: Solving the System

First, solve the third equation for (A): [A frac{1}{2}]

Substitute (A) into the first equation to find (B C): [frac{1}{2} B C 0 implies B C -frac{1}{2}]

Substitute (A) into the second equation to find (2B C): [-3left(frac{1}{2}right) - 2B - C 0 implies -frac{3}{2} - 2B - C 0 implies -2B - C frac{3}{2}]

Now we have a system of two linear equations with two variables: [begin{cases} B C -frac{1}{2} -2B - C frac{3}{2} end{cases}]

Solving this system, we can express (C) in terms of (B) in the first equation: [C -frac{1}{2} - B]

Substitute this expression for (C) into the second equation: [-2B - (-frac{1}{2} - B) frac{3}{2} implies -2B frac{1}{2} B frac{3}{2} implies -B frac{1}{2} frac{3}{2} implies B -1]

Now substitute (B) back to find (C): [C -frac{1}{2} - (-1) frac{1}{2}]

Thus, we have: [A frac{1}{2}, ; B -1, ; C frac{1}{2}]

Step 4: Writing the Partial Fraction Decomposition

Now we can write the partial fraction decomposition as:

[frac{1}{nn1n2} frac{1/2}{n} - frac{1}{n1} frac{1/2}{n2}]

Step 5: Summing the Series

Substituting back into the original series, we have:

[sum_{n1}^{infty} left(frac{1}{2}frac{1}{n} - frac{1}{n1} frac{1}{2}frac{1}{n2}right)]

This can be rewritten as:

[frac{1}{2} sum_{n1}^{infty} frac{1}{n} - sum_{n1}^{infty} frac{1}{n1} frac{1}{2} sum_{n1}^{infty} frac{1}{n2}]

Step 6: Evaluating the Series

Note that:

[sum_{n1}^{infty} frac{1}{n1} sum_{m2}^{infty} frac{1}{m}] [sum_{n1}^{infty} frac{1}{n2} sum_{m3}^{infty} frac{1}{m}]

Thus, we can express the series as:

[frac{1}{2} sum_{n1}^{infty} frac{1}{n} - left(sum_{n1}^{infty} frac{1}{n1} - frac{1}{2} sum_{n1}^{infty} frac{1}{n2}right)]

Since:

[sum_{n1}^{infty} frac{1}{n} - sum_{n1}^{infty} frac{1}{n1} 1] [frac{1}{2} sum_{n1}^{infty} frac{1}{n2} frac{1}{2} cdot left(-frac{1}{2}right) -frac{1}{4}]

Combining everything, we get:

[sum_{n1}^{infty} frac{1}{nn1n2} frac{1}{2} cdot 1 - 1 - frac{1}{2} cdot left(-frac{1}{2}right) frac{1}{2} - 1 frac{1}{4} frac{1}{4}]

Conclusion

We have thus proven that:

[boxed{sum_{n1}^{infty} frac{1}{nn1n2} frac{1}{4}}]