Question

If \(\left\\{a_{n}\right\\} \subset C\) and \(\lim a_{n}=a\), then \(\lim n^{-1} \sum_{1}^{n} a_{j}=a\).

Short answer

The sequence of averages \( b_n \) converges to \( a \).

Step-by-step solution

Understanding the Problem

We need to show that the sequence of averages, \( b_n = \frac{1}{n} \sum_{j=1}^{n} a_j \), converges to \( a \) as \( n \to \infty \). The sequence \( \{a_n\} \) is given to be a convergent sequence with limit \( a \).

Defining the Sequence of Averages

Define the sequence \( b_n = \frac{1}{n} \sum_{j=1}^{n} a_j \). We are asked to prove that \( \lim_{n \to \infty} b_n = a \).

Expressing Averages in Terms of Limits

Notice that \( b_n = \frac{1}{n}(a_1 + a_2 + \ldots + a_n) \). As each \( a_n \to a \), we expect the sum \( a_1 + a_2 + \ldots + a_n \) to approximate \( n \cdot a \) as \( n \) becomes large.

Using the Limit Properties

For large \( n \), most terms in \( \sum_{j=1}^{n} a_j \) will be close to \( a \). The sum is \( n \) times the average, so \( \frac{1}{n} \sum_{j=1}^{n} a_j \) approximates \( a \).

Applying the Limit Theorems

By the definition of convergence and properties of limits, the value \( b_n \to a \) can be shown rigorously using the formal definition of limits: for any \( \epsilon > 0 \), there exists an \( N \) such that for all \( n \geq N \), \( |a_n - a| < \epsilon \), ensuring \( |b_n - a| \) is also < \( \epsilon \).

Key concepts

Convergent Sequence

A convergent sequence is a sequence whose terms approach a specific value as the sequence progresses towards infinity. In mathematical terms, a sequence \( \{a_n\} \) is said to converge to a limit \( a \) if, for every positive number \( \epsilon \) (no matter how small), there exists a number \( N \) such that for all \( n \geq N \), the terms \( a_n \) are within an \( \epsilon \) distance from \( a \). This is formally written as \( \lim_{n \to \infty} a_n = a \).
To visualize, imagine plotting points on a number line. As you go further along in the sequence (i.e., as \( n \) becomes larger), the points get closer to the value \( a \), never straying more than \( \epsilon \) away from it, once you're past a certain position in the sequence.
This concept is foundational in calculus and analysis, and helps us to predict the behavior of functions and series over infinite intervals.

Sequence of Averages

The sequence of averages refers to the sequence formed by taking the average of the first \( n \) terms of another sequence \( \{a_n\} \). So, if you have a sequence \( \{a_n\} \), the sequence of averages \( \{b_n\} \) is defined by \( b_n = \frac{1}{n} \sum_{j=1}^{n} a_j \). Each term \( b_n \) represents the average value of the sequence \( \{a_n\} \) up to the \( n^{th} \) term.
This process is like a running average that smooths out the variations in \( \{a_n\} \), particularly useful when \( \{a_n\} \) is convergent. The sequence of averages can also be thought of as an indicator of long-term behavior and is important when examining consistency in patterns within sequences.

Limiting Behavior

Limiting behavior describes how the terms of a sequence behave as the index \( n \) becomes very large, approaching infinity. When we explore the limit of a sequence, we are concerned with what value the sequence tends to, if any, in its infinite journey.
In the context of the given problem, we examine the limiting behavior of the sequence of averages \( b_n = \frac{1}{n} \sum_{j=1}^{n} a_j \). Since each \( a_n \) in the sequence \( \{a_n\} \) approaches \( a \), the sequence of averages \( b_n \) also heads towards \( a \). This happens because the contribution of terms far back in the sequence becomes negligible, and more weight is put on terms closer to the limit as \( n \) increases.
Thus, despite potential fluctuations in earlier terms, the overarching trend moves towards convergence, which is precisely what limiting behavior encapsulates. This understanding is crucial for predicting long-term outcomes in mathematical sequences and series.