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Chapter 9: Problem 8

Let \(0

Short Answer

Expert verified
For the given series, the Root Test applies resulting in the series converging as the limit is less than 1, while the Ratio Test does not apply as the limit is inconclusive.

Step by step solution

01

Understanding the Test Rules

The Root Test requires the calculation of \(\lim_{n \to \infty} \sqrt[n]{|a_{n}|}\), whereas the Ratio Test requires \(\lim_{n \to \infty} |a_{n+1}/a_n|\). If either limit diverges or is not defined, then the respective test is not applicable.
02

Applying the Root Test

To apply the root test, calculate \(\lim_{n \to \infty} \sqrt[n]{|a^{2n}|}\) and \(\lim_{n \to \infty} \sqrt[n]{|a^{2n-1}|}\). Notice that as \(0<a<1\), the absolute value won't alter the values of these expressions. For \(a^{2n}\), the limit simplifies to \(\lim_{n \to \infty} {a^{2}}= a^{2}\), and for \(a^{2n-1}\), it simplifies to \(\lim_{n \to \infty} {a^{2}}= a^{2}\). Both of these limits are less than 1, so by the root test the series converges.
03

Root Test Done, Now Applying the Ratio Test

The Ratio Test works on the principle of ratios between consecutive terms. Thus, the limits to be considered are \(\lim_{n \to \infty} |a^{2n+1}/a^{2n}|\) and \(\lim_{n \to \infty} |a^{2(n+1)}/a^{2n-1}|\). On simplification, these become \(\lim_{n \to \infty} a\), and \(\lim_{n \to \infty} a^{3}\) respectively. As there is no \(n\) in these expressions, the Ratio Test is inconclusive, and hence does not apply to this series.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Root Test
The Root Test is a powerful tool for determining whether a series converges or diverges. It involves taking the nth root of the absolute value of the nth term of the series and then finding the limit as n approaches infinity. To apply the Root Test, you need to calculate:
\[ \lim_{n \to \infty} \sqrt[n]{|a_n|} \]
If this limit is less than 1, the series converges; if it is greater than 1, the series diverges; and if it equals 1, the test is inconclusive.
In the exercise provided, the series consists of alternating terms of the form \( a^{2n} \) and \( a^{2n-1} \) with \( 0
Ratio Test
The Ratio Test is another criterion used to determine the convergence of series, particularly useful when terms involve factorials or exponential functions. It compares the size of consecutive terms by calculating the limit of their ratio as n approaches infinity:
\[ \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| \]
The series converges if this limit is less than 1, and diverges if it's greater than 1. If the limit equals 1, the test gives no information, and the series may converge or diverge.
In our exercise, when we try to apply the Ratio Test to the series \( a^{2}+a+a^{4}+a^{3}+\cdots \), we find that it does not yield a definitive conclusion. The limit calculations \( \lim_{n \to \infty} |a^{2n+1}/a^{2n}| \) and \( \lim_{n \to \infty} |a^{2(n+1)}/a^{2n-1}| \) give us values of \( a \) and \( a^3 \), respectively, neither of which depends on n. Since \( 0
Series Convergence
Understanding series convergence is crucial for mathematical analysis and its applications. A series is said to be convergent if the sum of its terms approaches a specific value as more terms are added; otherwise, it is divergent. There are several tests to determine series convergence, each with its specific conditions and scenarios where it is most effective.
When evaluating the convergence of a series like \( a^{2}+a+a^{4}+a^{3}+\cdots \), it's important to consider the nature of its terms. In our example, the series consists of powers of a real number \( a \) where \( 0
Limit of a Sequence
The concept of the limit of a sequence is foundational in calculus. It describes the value that a sequence approaches as the index (n) goes to infinity. A limit is defined by the expression:
\[ \lim_{n \to \infty} a_n \]
If the sequence \( {a_n} \) gets arbitrarily close to a specific number L as n increases, then L is called the limit of the sequence. If such an L exists, the sequence is said to be convergent; otherwise, it is divergent.
In the context of our exercise, understanding the limit of a sequence helps in applying both the Root Test and the Ratio Test. For example, as n grows without bound in the terms \( a^{2n} \) and \( a^{2n-1} \), the impact of the exponent becomes negligible, leading to the limit of the sequence in each case being simply \( a^2 \), which is less than 1. This is why the Root Test is a fitting choice for this particular series as it relies on the concept of limits to determine convergence.

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Most popular questions from this chapter

Consider the series $$1-\frac{1}{2}-\frac{1}{3}+\frac{1}{4}+\frac{1}{5}-\frac{1}{6}-\frac{1}{7}++--\cdots$$ where the signs come in pairs. Does it converge?

Show that if a series is conditionally convergent, then the series obtained from its positive terms is divergent, and the series obtained from its negative terms is divergent.

Show that if \(|x|<1\), then \(\operatorname{Arcsin} x=\sum_{n=0}^{\infty} \frac{1 \cdot 3 \cdots(2 n-1)}{2 \cdot 4 \cdots 2 n} \cdot \frac{x^{2 n+1}}{2 n+1}\).

Discuss the convergence and the uniform convergence of the series \(\sum f_{n}\), where \(f_{n}(x)\) is given by: (a) \(\left(x^{2}+n^{2}\right)^{-1}\), (b) \((n x)^{-2} \quad(x \neq 0)\), (c) \(\sin \left(x / n^{2}\right)\), (d) \(\left(x^{n}+1\right)^{-1} \quad(x \neq 0)\). (e) \(x^{n} /\left(x^{n}+1\right) \quad(x \geq 0)\), (f) \((-1)^{n}(n+x)^{-1} \quad(x \geq 0)\).

Show that if the partial sums \(s_{n}\) of the series \(\sum_{k=1}^{\infty} a_{k}\) satisfy \(\left|s_{n}\right| \leq M n^{r}\) for some \(r<1\), then the series \(\sum_{n=1}^{N} a_{n} / n\) converges.
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