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Magazine Chapter 11: Problem 50
Prove that the series $$\sum_{n=1}^{\infty} \frac{1}{1+2+3+\cdots+n}$$ converges.
Short Answer
Expert verified
The series converges by telescoping and simplifies to 2.
Step by step solution
01
Identify the Series
The series given is \( \sum_{n=1}^{\infty} \frac{1}{1+2+3+\cdots+n} \). It is an infinite series where each term is of the form \( \frac{1}{1+2+3+\cdots+n} \). The sequence \( 1+2+3+\cdots+n \) is the sum of the first \( n \) natural numbers.
02
Simplify the Denominator
The sum of the first \( n \) natural numbers is given by the formula \( \frac{n(n+1)}{2} \). Thus, each term in the series can be written as \( \frac{1}{\frac{n(n+1)}{2}} = \frac{2}{n(n+1)} \). So, our series becomes \( \sum_{n=1}^{\infty} \frac{2}{n(n+1)} \).
03
Partial Fraction Decomposition
The term \( \frac{2}{n(n+1)} \) can be decomposed using partial fractions. We can write it as \( \frac{2}{n(n+1)} = \frac{A}{n} + \frac{B}{n+1} \). Solving for \( A \) and \( B \) by equating numerators gives \( 2 = A(n+1) + Bn \). Expanding and comparing coefficients, we find \( A = 2 \) and \( B = -2 \). Thus, \( \frac{2}{n(n+1)} = \frac{2}{n} - \frac{2}{n+1} \).
04
Express the Series as a Telescoping Series
By partial fraction decomposition, the series \( \sum_{n=1}^{\infty} \left( \frac{2}{n} - \frac{2}{n+1} \right) \) becomes telescoping. In a telescoping series, many terms cancel each other out. Writing the first few terms: \((\frac{2}{1} - \frac{2}{2}) + (\frac{2}{2} - \frac{2}{3}) + (\frac{2}{3} - \frac{2}{4}) + \cdots\).
05
Examine the Cancellation Process
In this telescoping series, each positive term cancels with the subsequent negative term. Only the first term, \( \frac{2}{1} \), does not have a matching negative counterpart, and due to the series continuing to infinity, only a constant remains.
06
Conclude Convergence
Since almost all terms in the telescoping series cancel out, what remains is a finite sum that starts from the first term, \( \frac{2}{1} \). Hence, the series converges to \( 2 \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Telescoping Series
A telescoping series is a special type of series where most terms cancel out, leaving only a few terms behind. This is an intriguing property used to find the sum of an infinite series efficiently. In the example given, the series became telescoping through partial fraction decomposition: - Each term in the series is rewritten, causing successive terms to negate or "telescope" out parts of one another. - Observe how series like \( \sum_{n=1}^{\infty} \left( \frac{2}{n} - \frac{2}{n+1} \right) \) simplify greatly to just a simple number after cancellation. At the core, a telescoping series shows a rapid progression towards its limit because the internal structure allows for cancellation. These series convert a seemingly complex problem into a simpler one, making them easy to evaluate.
Partial Fraction Decomposition
Partial fraction decomposition transforms a complex fraction into a sum of simpler fractions, making them easier to handle. It's a crucial technique in calculus, often used for integrals and series. Here's how it works: - The expression \( \frac{2}{n(n+1)} \) is decomposed into simpler terms \( \frac{2}{n} - \frac{2}{n+1} \). - This decomposition breaks the fraction into parts that can be handled individually, leading to simplification.The process involves equating coefficients:- Set up the fraction \( \frac{2}{n(n+1)} = \frac{A}{n} + \frac{B}{n+1} \). - Find values for \( A \) and \( B \) by solving equations resulting from comparing coefficients.In this way, we achieve a format that simplifies calculations, leading into telescoping series for the given exercise. This approach can simplify many complex mathematical tasks.
Finite Sum
A finite sum is what remains when an infinite series cancels out most of its terms, as seen in a telescoping series. Despite being derived from an infinite series, the concept of a finite sum assures us that the series has a limit or a specific value it approaches as the number of terms increases indefinitely. For our series:- While the series appears infinite, using the telescoping process, it effectively reduces to the value \( 2 \). - This occurs because all terms except the very first one get cancelled out.This result provides a concrete number, confirming that the series doesn't simply continue forever without reaching a specific value. In practical terms, understanding a finite sum gives us a tangible outcome from what initially seems to be an unmanageable problem.
