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Chapter 1: Problem 6

Test the following series for convergence. $$\sum_{n=1}^{\infty} \frac{(-1)^{n} n}{n+5}$$

Short Answer

Expert verified
The series ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline

Step by step solution

01

Identify the Series Type

The given series is ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline
02

Title

The given series is ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline

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Key Concepts

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

Alternating Series
An alternating series is a series where the terms alternate in sign. This means the terms switch from positive to negative, or negative to positive. For example, the series ewlineewlineewlineewlineewlineewlineewlineewlineewlineewlineewlineewlineewlineewlineewlineewlineewlineewlineewlineewline ewlineewline is an alternating series. In the given exercise, the series ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline does this because of the factor ewline ewline ewline, which makes the terms flip between positive and negative. Alternating series often need special convergence tests since their behavior can be tricky to analyze.
Convergence Tests
To determine whether an infinite series converges or diverges, we use convergence tests. In the context of alternating series, we often use the Alternating Series Test (AST). The AST states that an alternating series (with terms of the form ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline) converges if the absolute value of the terms satisfies the following properties: ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline Because ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ).
Infinite Series
An infinite series is the sum of the terms of an infinite sequence. This means as we add more terms, the series will infinitely grow or change. For example, ewlineewlineewlineewlineewlineewlineewlineewlineewlineewlineewlinenewlineewlineewlineewlineewlineewlineewlineewlineewline In the case of our exercise, the series ewlineewline ewline ewline yone term, it will remain infinite. so to determine the convergence, we analyze the behavior of partial sums (the sum of the first N terms) as N grows larger. If the partial sums approach a finite limit, we say the infinite series converges. If not, it diverges.
Mathematical Proofs
A mathematical proof is a logical argument demonstrating that a specific statement is true. Proofs use previously established results and logical reasoning. In analyzing series convergence, proofs often involve using various tests (like the AST or Comparison Test) and logical deductions. ewline ewline ewline ewline ewline ewline ewlineline ewline ewline Because ewline ewline ewline ewline)

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Most popular questions from this chapter

Find the interval of convergence of each of the following power series; be sure to investigate the endpoints of the interval in each case. $$\sum_{n=1}^{\infty} \frac{(-1)^{n} x^{n}}{\sqrt{n}}$$

Test the following series for convergence or divergence. Decide for yourself which test is easiest to use, but don't forget the preliminary test. Use the facts stated above when they apply. $$\sum_{n=1}^{\infty} \frac{(-1)^{n}}{2^{\ln n}}$$

Test the following series for convergence or divergence. Decide for yourself which test is easiest to use, but don't forget the preliminary test. Use the facts stated above when they apply. $$\sum_{n=0}^{\infty} \frac{(2 n) !}{3^{n}(n !)^{2}}$$

Use the special comparison test to find whether the following series converge or diverge. $$\sum_{n=0}^{\infty} \frac{n(n+1)}{(n+2)^{2}(n+3)}$$

Prove that an absolutely convergent series \(\sum_{n=1}^{\infty} a_{n}\) is convergent. Hint: Put \(b_{n}=\) \(a_{n}+\left|a_{n}\right| .\) Then the \(b_{n}\) are nonnegative; we have \(\left|b_{n}\right| \leq 2\left|a_{n}\right|\) and \(a_{n}=b_{n}-\left|a_{n}\right|.\)
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