Problem 72
Question
State the \(n\) th-Term Test for Divergence.
Step-by-Step Solution
Verified Answer
The nth-Term Test for Divergence states that if the limit as \(n\) approaches infinity of the \(n\)th term of a series does not equal zero, the series is divergent.
1Step 1: Understanding the nth-Term Test for Divergence
The nth-term test is a basic tool in calculus used to test the convergence or divergence of a series. It's straightforward and easy to apply. The big focus is on calculating the limit of the sequence forming the series.
2Step 2: Statement of the nth-Term Test for Divergence
The nth-Term Test for Divergence states: For a given series, if the limit as \(n\) approaches infinity of the \(n\)th term does not equal zero, then the series diverges. In mathematical terms, this is expressed as: If \(\lim_{n \to \infty} a_n \neq 0\), then the series \(\sum_{n=1}^\infty a_n\) is divergent.
3Step 3: Applications of the nth-Term Test for Divergence
It's important to note that if the \(n\)th term does go to zero, this test doesn't tell us anything about whether the series converges or diverges. For that, other series tests like the Integral Test, Comparison Test, or the Ratio Test might be utilized.
Other exercises in this chapter
Problem 71
In Exercises 71-74, evaluate the binomial coefficient using the formula \(\left(\begin{array}{l}k \\ n\end{array}\right)=\frac{k(k-1)(k-2)(k-3) \cdot \cdots(k-n
View solution Problem 71
Use the Root Test to determine the convergence or divergence of the series. $$ \sum_{n=1}^{\infty}\left(\frac{1}{n}-\frac{1}{n^{2}}\right)^{n} $$
View solution Problem 72
Determine whether the sequence with th given \(n\) th term is monotonic. Discuss the boundedness of th sequence. Use a graphing utility to confirm your results.
View solution Problem 72
Prove that if \(P(n)\) and \(Q(n)\) are polynomials of degree \(j\) and \(k\) respectively, then the series \(\sum_{n=1}^{\infty} \frac{P(n)}{Q(n)}\) converges
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