Question

Is the series convergent or divergent? $$ \sum_{n=1}^{\infty} \frac{1}{n^{2}+5 n+4} $$

Short answer

The series is convergent.

Step-by-step solution

Recognize the Type of Series

The series is given as \( \sum_{n=1}^{\infty} \frac{1}{n^{2} + 5n + 4} \). This is an infinite series with terms in the form of a rational function. To determine convergence, we'll apply suitable convergence tests for such forms.

Simplify the Denominator If Possible

The denominator \( n^2 + 5n + 4 \) can be factored as \( (n+1)(n+4) \). This simplifies each term of the series to \( \frac{1}{(n+1)(n+4)} \). This indicates that the series might be a telescoping series.

Express as Partial Fractions

To confirm if the series is telescoping, express \( \frac{1}{(n+1)(n+4)} \) using partial fraction decomposition: \( \frac{1}{(n+1)(n+4)} = \frac{A}{n+1} + \frac{B}{n+4} \). Solve for \( A \) and \( B \) by equating coefficients after simplification.

Solve for A and B

Setting \( \frac{1}{(n+1)(n+4)} = \frac{A}{n+1} + \frac{B}{n+4} \), we multiply through by the common denominator to get \( 1 = A(n+4) + B(n+1) \). Expanding gives \( 1 = (A+B)n + (4A+B) \). Solving the system \( A + B = 0 \) and \( 4A + B = 1 \) yields \( A = 1/3 \), \( B = -1/3 \).

Write the Series in Telescoping Form

Substitute \( A \) and \( B \) back into the fractions: \( \frac{1}{(n+1)(n+4)} = \frac{1}{3} \frac{1}{n+1} - \frac{1}{3} \frac{1}{n+4} \). The series \( \sum_{n=1}^{\infty} \left( \frac{1}{3(n+1)} - \frac{1}{3(n+4)} \right) \) is a telescoping series.

Analyze the Telescoping Nature

In a telescoping series, most terms cancel out. Specifically, in the partial sums \( S_N = \sum_{n=1}^{N} \left( \frac{1}{3(n+1)} - \frac{1}{3(n+4)} \right) \), everything that doesn't cancel will tend towards a limit as \( N \to \infty \).

Determine Convergence

Calculate the first few terms and note how terms cancel: \( S_N = \frac{1}{3} \left( \frac{1}{2} + \frac{1}{3} + \frac{1}{4} - \frac{1}{N+2} - \frac{1}{N+3} - \frac{1}{N+4} \right) \). As \( N \to \infty \), remaining terms in the sum tend towards finite values. Thus, the series converges.

Key concepts

Telescoping Series

A telescoping series is a special kind of series where many terms cancel out, making it easier to find the sum or convergence of the series. This property typically arises after expressing the terms in a specific form, allowing terms from consecutive partial sums to subtract each other out.

In the context of the exercise, the series can be rewritten using partial fractions, revealing its telescopic nature. Specifically, for the given series, each term can be expressed in a way that the terms overlap and cancel, except for a few terms at the ends of the series. This dramatic cancellation simplifies the series to a few initial and terminal terms as more and more terms are added. Ultimately, this makes it possible to determine if the entire series converges by calculating the limit of the remaining terms as the number of terms approaches infinity.

Partial Fraction Decomposition

Partial fraction decomposition is a technique used to break down complex rational expressions into simpler fractions, which makes it easier to add, subtract, or apply other operations on them. This is particularly useful when dealing with rational functions in the context of series.

For the infinite series exercise, the rational function \( \frac{1}{n^2 + 5n + 4} \) was transformed into partial fractions, making it easier to identify its telescoping nature. By expressing the fraction as \( \frac{1}{n+1} - \frac{1}{n+4} \), we turned it into a form that exposes the canceling pattern inherent in the series. This not only simplifies calculations but also directly assists in determining convergence by visually revealing the canceling terms.

Infinite Series

An infinite series is a summation of terms that continue indefinitely. It is a central concept in calculus and analysis where we investigate whether an endless accumulation of numbers approaches a finite limit or continues to grow without bound.

In this exercise, we examine the infinite series given by \( \sum_{n=1}^{\infty} \frac{1}{n^2 + 5n + 4} \). Through simplification and transformation, understanding its behavior becomes manageable, allowing us to establish if it converges to a finite limit. This kind of analysis is crucial in many scientific and engineering applications, where long-term behavior or predictions are based on the aggregation of numerous inputs calculated through infinite processes.

Rational Functions

Rational functions are expressed as the ratio of two polynomials. These functions can model countless phenomena in math and science due to their versatile form, making them a pervasive element in calculus problems, including those involving series.

In the context of the given problem, the term \( \frac{1}{n^2 + 5n + 4} \) is a rational function where both the numerator and the denominator are polynomials. Factoring the denominator and applying partial fraction decomposition help simplify the problem significantly. Understanding these functions not only aids in solving series but also provides insight into their behavior and properties. For students, mastering rational functions is key to successful series convergence analysis.