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Questions and Answers
What can be inferred about the limit as $n$ approaches infinity of the expression $\frac{(-1)^n}{n+1}$?
What can be inferred about the limit as $n$ approaches infinity of the expression $\frac{(-1)^n}{n+1}$?
- The limit equals 0 (correct)
- The limit diverges to infinity
- The limit oscillates between -1 and 1
- The limit equals 1
Which of the following statements about series convergence is true based on the limit expression?
Which of the following statements about series convergence is true based on the limit expression?
- The convergence cannot be determined from the limit alone
- The series converges due to the nature of the limit (correct)
- The value of the limit indicates a finite sum
- The series diverges because it oscillates
In the expression $\lim_{n\to\infty} \frac{(-1)^n}{n+1}$, what role does the term $(-1)^n$ play?
In the expression $\lim_{n\to\infty} \frac{(-1)^n}{n+1}$, what role does the term $(-1)^n$ play?
- It oscillates the terms of the series (correct)
- It stabilizes the growth of the series
- It influences the limit's value directly
- It causes the limit to diverge
What is the significance of the term $n + 1$ in the denominator of the limit expression?
What is the significance of the term $n + 1$ in the denominator of the limit expression?
What does the expression $\lim_{n\to\infty} \frac{(-1)^n}{n+1} = 0$ imply about the behavior of the series defined by the terms $(-1)^n$?
What does the expression $\lim_{n\to\infty} \frac{(-1)^n}{n+1} = 0$ imply about the behavior of the series defined by the terms $(-1)^n$?
Study Notes
Limits and Convergence
- The expression lim (-1)^n / (n+1) = 0 as n approaches infinity suggests an exploration of a sequence's behavior as the number of terms increases.
- The notation "lim" signifies taking the limit of a sequence, which essentially describes the long-term behavior of its values.
- The expression (-1)^n represents an alternating sequence, where the terms alternate between positive and negative values.
- Dividing (-1)^n by (n+1) introduces a factor that decreases as n grows larger, effectively "dampening" the alternating behavior.
- The limit of 0 implies that, as n becomes infinitely large, the terms of the sequence (-1)^n / (n+1) get arbitrarily close to zero.
- This scenario likely indicates that the associated series converges, meaning the sum of its infinite terms approaches a finite value.
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Description
Explore the fascinating topic of limits and convergence in sequences, focusing on the expression lim (-1)^n / (n+1) as n approaches infinity. Understand how alternating sequences behave and how their limits reveal important properties about series convergence. Dive into the mathematics that govern long-term behavior and finite sums.
