Question

Here are some telescoping series problems: a. Verify that $$\sum _{n=1}^{\mathrm{\infty }}\frac{1}{(n+2)(n+1)}=\sum _{n=1}^{\mathrm{\infty }}(\frac{n+1}{n+2}-\frac{n}{n+1})$$ b. Find the $n$th partial sum of the series $\sum _{n=1}^{\mathrm{\infty }}(\frac{n+1}{n+2}-\frac{n}{n+1})$ and use it to determine the sum of the resulting telescoping series. c. Sum the series $\sum _{n=1}^{\mathrm{\infty }}[{\tan }^{-1}n-{\tan }^{-1}(n+1)]$ by first writing the $N$ th partial sum and then computing $\underset{N\to \mathrm{\infty }}{lim}{s}_N$

Short answer

a) The identity is verified by writing out the first few terms and observing the cancelations. b) The sum of the series is 1. c) The sum of the series is $\frac{\pi }{4}$.

Step-by-step solution

Part (a) Verification of the Identity

Write out a few terms of the sum to verify the identity. Cancellation of terms will occur, and the proof of the identity relies on these cancellations.

Part (b) Find the nth Partial Sum

To find the nth partial sum (denoted $S_n$), write the sum and withdraw terms until a pattern appears. From there, generalize $S_n$ in terms of n.

Part (b) Find the Sum of the Series

To find the sum of the series, compute $\underset{n\to \mathrm{\infty }}{lim}{S}_n$. The sum of the series, if it exists, is this limit.

Part (c) Rewrite the Series

Write the series as the partial sum, $S_N$, then compute $\underset{N\to \mathrm{\infty }}{lim}{S}_N$. The sum of this series, if it exists, is this limit.

Key concepts

Partial Sums

When working with series, a partial sum, denoted as $S_n$, is the sum of the first $n$ terms of the series. It acts as a crucial step to understanding the behavior of an infinite series. In essence, it provides insight into how the series progresses and whether or not it will converge. By examining the partial sums, patterns may emerge that are not obvious from the original series.

To find a partial sum, you'll begin by writing out a set number of terms from the series. Sometimes, especially in telescoping series, you will notice that many terms cancel out. This simplification allows us to identify a pattern or formula for $S_n$.

For instance, in the given series $\sum _{n=1}^{\mathrm{\infty }}(\frac{n+1}{n+2}-\frac{n}{n+1})$, the challenge is to express $S_n$, the partial sum of the first $n$ terms, in such a way that makes it easy to see its overall behavior. This pattern in telescoping series is often simplified down to a couple of terms as the middle terms cancel, giving us a clean final expression.

Limit of a Series

The limit of a series is what the partial sums $S_n$ approach as $n$ grows larger and larger, i.e., approaches infinity. Finding the limit is essential as it determines the convergence or divergence of the series. If the partial sum $S_n$ tends to a specific finite number as $n\to \mathrm{\infty }$, the series converges, and that finite number is the sum of the series. Otherwise, the series diverges.

In practice, this means calculating $\underset{n\to \mathrm{\infty }}{lim}{S}_n$. In the context of telescoping series like $\sum _{n=1}^{\mathrm{\infty }}(\frac{n+1}{n+2}-\frac{n}{n+1})$, after revealing the simplified form of the $n$-th partial sum through cancellations, you perform this limit process. The final expression might become more straightforward to evaluate as $n$ grows very large, often revealing constants or zero as the result.

Series Convergence

Understanding whether a series converges or diverges is a central concept in analyzing infinite series. Convergence means that as we add more terms from the series, the partial sum becomes closer and closer to a certain value. Telescoping series often simplify directly to a convergent series, owing to term cancellations.

Criteria and tests for convergence are numerous in series analysis. However, specific to the telescoping series, convergence is typically managed by writing the series in a form where consecutive terms cancel each other out, simplifying the series significantly. This technique makes it evident whether the series reaches a particular number or continues indefinitely.

For example, with the series $\sum _{n=1}^{\mathrm{\infty }}[{\tan }^{-1}n-{\tan }^{-1}(n+1)]$, rewriting it could have significant term cancellations making it clear whether a sum exists and what the sum is if the series converges. Thus, by evaluating the partial sums and applying the limit process, we can determine the series' convergence and identify its sum if one exists.