Suppose an and bn are sequences with positive terms, and the series ∑bn is known to be convergent. Which of the following statements is true?

If lim ⁡ n → ∞ ⁡ a n b n = 0 , then ∑ a n converges.

Suppose a n and b n are sequences with positive terms, and the series ∑ b n is known to be convergent. Which of the following statements is true?

If lim ⁡ n → ∞ ⁡ a n b n = 0 , then ∑ a n converges.

If lim ⁡ n → ∞ ⁡ a n b n = 0 , then ∑ a n diverges.

If lim ⁡ n → ∞ ⁡ a n b n = 1 , then ∑ a n diverges.

If lim ⁡ n → ∞ ⁡ a n b n = ∞ , then ∑ a n converges.

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0. Functions7h 52m
1. Limits and Continuity2h 2m
2. Intro to Derivatives1h 33m
3. Techniques of Differentiation3h 18m
4. Applications of Derivatives2h 38m
5. Graphical Applications of Derivatives6h 2m
6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m
7. Antiderivatives & Indefinite Integrals1h 26m
8. Definite Integrals4h 44m
9. Graphical Applications of Integrals2h 27m
- Area Between Curves
  1h 18m
- Introduction to Volume & Disk Method
  1h 8m
10. Physics Applications of Integrals 3h 16m
- Kinematics
  1h 13m
- Work
  2h 3m
11. Integrals of Inverse, Exponential, & Logarithmic Functions2h 34m
12. Techniques of Integration7h 39m
13. Intro to Differential Equations2h 55m
14. Sequences & Series5h 36m
15. Power Series2h 19m
- Introduction to Power Series
  1h 9m
- Taylor Series & Taylor Polynomials
  1h 10m
16. Parametric Equations & Polar Coordinates7h 58m

1. Limits and Continuity

Introduction to Limits

Calculus

1. Limits and Continuity

Introduction to Limits

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Multiple Choice

Suppose $a_{n}$ and $b_{n}$ are sequences with positive terms, and the series $\sum b_{n}$ is known to be convergent. Which of the following statements is true?

lim n \to \infty \frac{a_{n}}{b_{n}} = 0

, then

\sum a_{n}

diverges.

lim n \to \infty \frac{a_{n}}{b_{n}} = 0

, then

\sum a_{n}

converges.

lim n \to \infty \frac{a_{n}}{b_{n}} = 1

, then

\sum a_{n}

diverges.

lim n \to \infty \frac{a_{n}}{b_{n}} = \infty

, then

\sum a_{n}

converges.

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Verified step by step guidance

Step 1: Begin by understanding the comparison test and limit comparison test for series. These tests are used to determine the convergence or divergence of a series by comparing it to another series whose behavior is already known.

Step 2: Recall that if $ \sum b_n $ is a convergent series with positive terms, and $ \lim_{n \to \infty} \frac{a_n}{b_n} $ exists and is finite, then the behavior of $ \sum a_n $ (convergence or divergence) will match that of $ \sum b_n $.

Step 3: Analyze the given statements: If $ \lim_{n \to \infty} \frac{a_n}{b_n} = 0 $, this implies that $ a_n $ becomes much smaller than $ b_n $ as $ n \to \infty $. Since $ \sum b_n $ converges, $ \sum a_n $ will also converge.

Step 4: Consider the second statement: If $ \lim_{n \to \infty} \frac{a_n}{b_n} = 1 $, this implies that $ a_n $ and $ b_n $ are asymptotically similar. Since $ \sum b_n $ converges, $ \sum a_n $ will also converge.

Step 5: Evaluate the third statement: If $ \lim_{n \to \infty} \frac{a_n}{b_n} = \infty $, this implies that $ a_n $ grows much faster than $ b_n $. In this case, $ \sum a_n $ will diverge because the terms $ a_n $ do not decrease sufficiently to satisfy the convergence criteria.