Problem 46 - Single Variable Calculus

Question

Show that if \(\sum_{n=1}^{\infty} a_{n}\) is absolutely convergent, then \(\left|\sum_{n=1}^{\infty} a_{n}\right| \leq \Sigma_{n=1}^{\infty}\left|a_{n}\right|\)

Step-by-Step Solution

Answer

Using the triangle inequality, we can show that the absolute value of the sum of a convergent series is bounded by the sum of the absolute values of its terms. For partial sums, we have \( \left|\sum_{n=1}^{N} a_n\right| \leq \sum_{n=1}^{N} |a_n|\). Taking the limit as \(N\) goes to infinity, we get \(\left|\sum_{n=1}^{\infty} a_n\right| \leq \sum_{n=1}^{\infty} |a_n|\).

Step 1: Definition of absolute convergence

A series \(\sum_{n=1}^{\infty} a_n\) is absolutely convergent if the series \(\sum_{n=1}^{\infty} |a_n|\) is convergent. In other words, given that the series is absolutely convergent, we know that \(\sum_{n=1}^{\infty} |a_n| < \infty\).

Step 2: Triangle Inequality

Recall the triangle inequality: for any real numbers \(x\) and \(y\), we have that \(|x + y| \leq |x| + |y|\). In the context of our series, the triangle inequality tells us that $$ |a_n + a_{n+1}| \leq |a_n| + |a_{n+1}|. $$

Step 3: Apply Triangle Inequality to Partial Sums

We want to prove that \(\left|\sum_{n=1}^{\infty} a_n\right| \leq \sum_{n=1}^{\infty}\left|a_n\right|\). We will do this by showing that this is true for all partial sums up to a term \(N\). From the triangle inequality, we have that $$ \left|\sum_{n=1}^{N} a_n\right| \leq \sum_{n=1}^{N} |a_n|. $$ Since this is true for all partial sums up to \(N\), we can take the limit as \(N\) goes to infinity.

Step 4: Take Limit of Inequality

Taking the limit on both sides of the inequality as \(N\) goes to infinity, we have that $$ \lim_{N \to \infty} \left|\sum_{n=1}^{N} a_n\right| \leq \lim_{N \to \infty} \sum_{n=1}^{N} |a_n|. $$

Step 5: Replace Limit with Infinite Sums

Since the limit of partial sums is just the sum of the infinite series, we can rewrite the inequality as $$ \left|\sum_{n=1}^{\infty} a_n\right| \leq \sum_{n=1}^{\infty} |a_n|, $$ which is the result that we wanted to prove. In conclusion, if a series \(\sum_{n=1}^{\infty} a_n\) is absolutely convergent, its absolute value is less than or equal to the sum of its absolute values.

Key Concepts

triangle inequality

The triangle inequality is a fundamental concept in mathematics, particularly in analysis and geometry. It provides a way to compare the size of a sum to the size of its parts. Specifically, for any real numbers or vectors, the triangle inequality states: \(|x + y| \leq |x| + |y|\).

This inequality implies that the absolute value of a sum is always less than or equal to the sum of the absolute values. It's a useful tool when dealing with series and sequences.

• For any two elements: It helps in understanding how the sum behaves in terms of magnitude.
• In series: Allows us to apply this concept to partial sums \(|a_n + a_{n+1}| \leq |a_n| + |a_{n+1}|\).

By applying the triangle inequality repeatedly to the terms of a series, we can make broader statements about the series' behavior, leading to conclusions about convergence and limits.

limit of partial sums

In the study of series, partial sums play a crucial role in understanding convergence. A partial sum is simply the sum of the first \(N\) terms of a series, expressed as \(S_N = \sum_{n=1}^{N} a_n\). As \(N\) increases, these partial sums approach a certain value, called the limit of the series.

This concept is critical when determining whether a series converges. A series \(\sum_{n=1}^{\infty} a_n\) converges if the sequence of its partial sums \(S_N\) converges to a limit \(L\).

• Helps in proving convergence: If \(\lim_{N \to \infty} S_N = L\), the series converges to \(L\).
• Tool in inequalities: Used to take the limit on both sides of an inequality involving series or sequences.

By examining the limit of partial sums, we can ascertain the behavior of the series as \(N\) becomes very large, which is essential in many analysis problems.

infinite series

An infinite series is the sum of the terms of an infinite sequence, expressed as \(\sum_{n=1}^{\infty} a_n\). These series are a key concept in calculus and analysis, helping to explain various functions and mathematical phenomena.

Convergence of an infinite series means that as you add more and more terms, the total approaches a specific value. There are several tests and techniques to determine whether a series converges.

• Absolute convergence: If \(\sum_{n=1}^{\infty} |a_n|\) converges, the series \(\sum_{n=1}^{\infty} a_n\) is absolutely convergent, which generally ensures the series converges.
• Conditional convergence: Occurs if \(\sum_{n=1}^{\infty} a_n\) converges but \(\sum_{n=1}^{\infty} |a_n|\) does not.

Understanding infinite series is essential, as they are applied across different scientific disciplines, in solving differential equations, and in approximations of functions used in engineering and physics.