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Magazine Chapter 8: Problem 88
Prove that if \(0 \leq a_{n}\) and \(\sum_{n=1}^{\infty} a_{n}\) converges then
there exist numbers \(b_{n}\) with \(a_{n}
Short Answer
Expert verified
There exists \(b_n = a_n + \frac{1}{2^n}\) such that \(a_n < b_n\) and \(\sum_{n=1}^{\infty} b_n\) converges.
Step by step solution
01
Understanding the Problem
We need to find a sequence \(b_n\) such that \(a_n < b_n\) for every \(n\), and the sum \(\sum_{n=1}^{\infty} b_{n}\) is convergent while \(\sum_{n=1}^{\infty} a_{n}\) is already convergent.
02
Analyze the Convergent Series Property
Since \(\sum_{n=1}^{\infty} a_{n}\) is convergent, the sequence \(a_n\) must tend to zero as \(n\) approaches infinity. This is because a necessary condition for the convergence of an infinite series is that the terms must go to zero.
03
Construct the Sequence \(b_n\)
Define \(b_n = a_n + \frac{1}{2^n}\). Since \(0 \leq a_n\), we have \(a_n < a_n + \frac{1}{2^n} = b_n\). The sequence \(b_n\) incorporates the original sequence plus a shrinking positive value \(\frac{1}{2^n}\), which ensures that \(b_n > a_n\).
04
Prove Convergence of \(\sum_{n=1}^{\infty} b_n\)
Consider the series \(\sum_{n=1}^{\infty} b_n = \sum_{n=1}^{\infty} (a_n + \frac{1}{2^n})\). This can be split into two series: \(\sum_{n=1}^{\infty} a_n\) which is convergent by assumption, and \(\sum_{n=1}^{\infty} \frac{1}{2^n}\), which is a geometric series. The series \(\sum_{n=1}^{\infty} \frac{1}{2^n}\) converges because its starting term is \(\frac{1}{2}\) and the common ratio is \(\frac{1}{2}\), which is less than 1.
05
Conclude the Proof
The convergence of both \(\sum_{n=1}^{\infty} a_n\) and \(\sum_{n=1}^{\infty} \frac{1}{2^n}\) implies the convergence of their sum, \(\sum_{n=1}^{\infty} b_n\). Hence, we have found a sequence \(b_n\) such that \(a_n < b_n\) and \(\sum_{n=1}^{\infty} b_n\) converges.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Sequence Analysis
Sequence analysis is a fundamental concept in mathematics. This involves examining the behavior of sequences as their terms progress towards infinity. A sequence can be thought of as a list of numbers arranged in a specific order, typically denoted as \(a_1, a_2, a_3, \ldots \). The objective of sequence analysis is often to understand the long-term behavior of these numbers.
- Convergence of a Sequence: A sequence \(a_n\) is said to converge if, as \(n\) approaches infinity, \(a_n\) approaches a fixed number. This fixed number is called the limit.
- Monotonicity: If a sequence consistently increases or decreases, it is called monotonic. This can help in determining convergence.
- Boundedness: A sequence is bounded if there is a real number that exceeds or bounds each term of the sequence. Bounded and monotonic sequences are particularly important because they are guaranteed to converge.
Infinite Series
An infinite series is a sum of an infinite sequence of terms. We usually represent it in the form \(\sum_{n=1}^{\infty} a_n\). The concepts involved in infinite series are crucial because they help mathematicians and students understand how to approach sums that have no definite end. Let's look into a few key ideas:
- Convergence of a Series: An infinite series \(\sum_{n=1}^{\infty} a_n\) converges if the sequence of its partial sums \(s_n = a_1 + a_2 + \cdots + a_n\) approaches a finite limit as \(n\) goes to infinity.
- Partial Sums: These are used to find out whether an infinite series converges. If the sequence of partial sums reaches a certain value and stays there, the series is convergent.
- Divergence: If the sequence of partial sums does not approach a finite limit as \(n\) becomes large, the infinite series is said to diverge.
Geometric Series Convergence
A geometric series is a specific type of infinite series where each term is a constant multiple of the previous term. It has the general form \(a, ar, ar^2, ar^3, \ldots \). The convergence of a geometric series depends primarily on the common ratio \(r\).
- Convergence Condition: A geometric series converges if the absolute value of the common ratio \(|r|\) is less than 1. This is because the terms get progressively smaller, allowing the series to sum to a finite value.
- Sum Formula for Convergent Geometric Series: If \(|r| < 1\), the sum of the geometric series can be calculated as \( \sum_{n=0}^{\infty} ar^n = \frac{a}{1 - r} \).
- Example: The series \(\sum_{n=1}^{\infty} \frac{1}{2^n}\) is a geometric series with \(a = \frac{1}{2}\) and \(r = \frac{1}{2}\). Since \(|r| < 1\), it converges.
