Mastering Telescoping Series: Convergence and Divergence

Dive into the world of telescoping series formula. Learn to identify, simplify, and evaluate these unique series. Discover applications in calculus and beyond. Elevate your mathematical prowess today!

Convergence of Telescoping Series
Show that the following series are convergent and find its sum:
$\sum_{n=1}^{\infty}\frac{4}{n^2+7n+12}$

Step 1: Recognize the Series as Telescoping

We start by recognizing that the given series is a telescoping series. This is evident because the problem is situated in the context of telescoping series. The goal is to show that the series converges and to find its sum.

Step 2: Factor the Denominator

Next, we focus on the denominator of the series term $\frac{4}{n^2+7n+12}$. We need to factorize the quadratic expression in the denominator. The quadratic expression $n^2 + 7n + 12$ can be factored as $(n + 3)(n + 4)$. Therefore, the series term can be rewritten as:

$\frac{4}{(n+3)(n+4)}$

Step 3: Partial Fraction Decomposition

We then decompose the fraction into partial fractions. We express $\frac{4}{(n+3)(n+4)}$ as:

$\frac{4}{(n+3)(n+4)} = \frac{A}{n+3} + \frac{B}{n+4}$

To find the values of $A$ and $B$, we multiply both sides by $(n+3)(n+4)$ to clear the denominators:

$4 = A(n+4) + B(n+3)$

We then solve for $A$ and $B$ by choosing suitable values for $n$.

Step 4: Solve for A and B

First, let $n = -3$:

$4 = A(-3+4) + B(-3+3)$

$4 = A(1) + B(0)$

$A = 4$

Next, let $n = -4$:

$4 = A(-4+4) + B(-4+3)$

$4 = A(0) + B(-1)$

$B = -4$

Thus, we have $A = 4$ and $B = -4$. Substituting these values back into the partial fractions, we get:

$\frac{4}{(n+3)(n+4)} = \frac{4}{n+3} - \frac{4}{n+4}$

Step 5: Write the Series in Telescoping Form

We rewrite the original series using the partial fractions:

$\sum_{n=1}^{\infty} \left( \frac{4}{n+3} - \frac{4}{n+4} \right)$

We can now see that this is a telescoping series, where most terms will cancel out.

Step 6: Expand the Series and Identify Cancellation

We expand the series to observe the cancellation of terms:

$\left( \frac{4}{4} - \frac{4}{5} \right) + \left( \frac{4}{5} - \frac{4}{6} \right) + \left( \frac{4}{6} - \frac{4}{7} \right) + \cdots$

We notice that each term $\frac{4}{n+4}$ cancels with the subsequent term $\frac{4}{n+3}$, leaving only the first term of the series and the last term of the series.

Step 7: Evaluate the Remaining Terms

After cancellation, we are left with the first term and the last term of the series:

$1 - \frac{4}{N+4}$

where $N$ is the upper limit of the partial sum.

Step 8: Take the Limit as N Approaches Infinity

Finally, we take the limit as $N$ approaches infinity:

$\lim_{N \to \infty} \left( 1 - \frac{4}{N+4} \right) = 1 - 0 = 1$

Thus, the series converges, and the sum of the series is 1.

Here are some frequently asked questions about telescoping series:

1. How do you calculate telescoping series?

To calculate a telescoping series, follow these steps: 1. Identify the general term of the series. 2. Use partial fraction decomposition if necessary. 3. Write out the series and look for cancelling terms. 4. Sum the remaining terms after cancellation. 5. Evaluate the limit of the sum as n approaches infinity.

2. What is the formula for telescope series?

The general formula for a telescoping series is: Σ(n=a to b) (f(n) - f(n+1)) = f(a) - f(b+1) Where f(n) is some function of n, and a and b are the lower and upper bounds of the summation.

3. What is the telescopic series rule?

The telescopic series rule states that for a series of the form Σ(n=1 to ) (f(n) - f(n+1)), if the limit of f(n) as n approaches infinity exists and equals L, then the sum of the series is f(1) - L.

4. What is the telescoping sum theorem?

The telescoping sum theorem states that for a finite telescoping series Σ(n=1 to N) (a(n) - a(n+1)), the sum is equal to a(1) - a(N+1). This theorem is the basis for evaluating both finite and infinite telescoping series.

5. How do you identify a telescoping series?

To identify a telescoping series: 1. Look for terms that can be written as a difference of two functions. 2. Check if adjacent terms have parts that cancel when added. 3. Use partial fraction decomposition to reveal hidden telescoping patterns. 4. Verify if the series can be simplified to a difference between the first term and the limit of the last term.

Understanding the convergence and divergence of telescoping series is a crucial concept in advanced calculus. To fully grasp this topic, it's essential to have a solid foundation in several prerequisite areas. One of the most fundamental concepts is convergence and divergence of normal infinite series, which forms the basis for understanding more complex series types.

Building on this, familiarity with convergence and divergence of geometric series is crucial, as telescoping series often involve manipulating terms in ways similar to geometric series. This knowledge helps in recognizing patterns and applying appropriate convergence tests.

A strong grasp of partial fraction decomposition is invaluable when working with telescoping series. This technique allows for breaking down complex fractions into simpler terms, which is often a key step in identifying telescoping behavior.

Understanding the domain and range of a function is crucial when dealing with series, as it helps in determining the validity of certain operations and the behavior of the series at different points. This concept ties closely with limits at infinity and horizontal asymptotes, which are essential for analyzing the long-term behavior of telescoping series.

The divergence of harmonic series serves as an important counterexample and helps in developing intuition about when series might converge or diverge. This knowledge is particularly useful when dealing with telescoping series that resemble or can be compared to harmonic series.

While not directly related to telescoping series, understanding the method of undetermined coefficients can be helpful in manipulating series and finding general terms, which is often a crucial step in working with telescoping series.

Lastly, proficiency in simplifying complex fractions is indispensable when working with telescoping series. The ability to manipulate and simplify fractions efficiently often makes the difference in successfully identifying and summing telescoping series.

By mastering these prerequisite topics, students will be well-equipped to tackle the intricacies of convergence and divergence in telescoping series. Each concept builds upon the others, creating a comprehensive understanding that allows for deeper insights and more effective problem-solving in this advanced area of calculus.

• Partial fraction decomposition
• Introduction to infinite series