题目

MAT137Y1 LEC 20249: Calculus with Proofs (all lecture sections) Pre-Class Quiz 61(13.13 and 13.14)

多项选择题

Let 𝑓 be a continuous function with domain 𝑅 . Assume 𝑓 is POSITIVE and DECREASING. Assume lim 𝑥 → ∞ 𝑓 ( 𝑥 ) = 0 . Which of the following statements MUST be true? Select all the correct answers.

选项

A.∑ 𝑛 = 1 ∞ ( − 1 ) 𝑛 𝑓 ( 𝑛 ) is divergent.

B.IF ∫ 100 ∞ 𝑓 ( 𝑥 ) 𝑑 𝑥 is convergent, THEN ∑ 𝑛 = 1 ∞ 𝑓 ( 𝑛 ) is convergent.

C.∑ 𝑛 = 1 ∞ ( − 1 ) 𝑛 𝑓 ( 𝑛 ) is convergent.

D.IF ∑ 𝑛 = 1 ∞ 𝑓 ( 𝑛 ) is convergent, THEN ∫ 1 ∞ 𝑓 ( 𝑥 ) 𝑑 𝑥 is convergent.

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思路分析

We are given a continuous, positive, decreasing function f on the real line with limit zero as x → ∞. We will analyze each statement in light of standard convergence tests. Option 1: ∑_{n=1}^∞ (-1)^n f(n) is convergent. - Because f is positive, decreasing, and tends to 0, the alternating series test (Leibniz criterion) applies: the terms f(n) decrease to 0, so the alternating sum ∑ (-1)^......Login to view full explanation

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类似问题

Question textFor each of the series below you find two answer fields. In the first answer field enter: (inputs are case sensitive) [table] DV | | if the series is divergent and not equal to ±∞±∞\pm\infty CV | | if the series is convergent (to a non-zero number) Z | | if the series converges to 0 INF | | if the series equals to ∞∞ \infty NIF | | if the series equals to −∞−∞ -\infty [/table] [table] WD | | if the series is not well defined | | [/table] In the second answer field select one of the following reasons that can be used to prove your claim in the first answer field: [table] DT | | The Divergency Test IT | | The Integral Test AS | | The Alternating Series Test RO | | The Root Test RA | | The Ratio Test [/table] [table] D | | The sequence of summands decreases to 00 0 L | | The limit of summands exists and equals to 00 0 C | | Comparison with a geometric series ∑∞𝑛=0𝑞𝑛∑n=0∞qn \sum_{n=0}^{\infty}q^n CH | | Comparison with the harmonic series AH | | Comparison with the alternating harmonic series [/table] [table] P | | Comparison with 𝑝pp-series, where 𝑝>1p>1p > 1 LP | | Comparison with 𝑝pp-series, where 𝑝<1p<1p < 1 [/table] [table] ∑𝑛=1∞(𝑛+1)7(8⋅𝑛!)2(2𝑛)!∑n=1∞(n+1)7(8⋅n!)2(2n)! \sum_{n=1}^\infty (n+1)^{{7}} \,\, \frac{({8}\cdot n!)^2 }{ (2n)! } | | because ∑𝑛=1∞(1−6𝑛)2𝑛2∑n=1∞(1−6n)2n2 \sum\limits_{n=1}^{\infty} \left(1- \frac{{6} }{ n } \right)^{ {2}n^2} | | because [/table]

Question textFor each of the series below you find two answer fields. In the first answer field enter: (inputs are case sensitive) [table] DV | | if the series is divergent and not equal to ±∞±∞\pm\infty CV | | if the series is convergent (to a non-zero number) Z | | if the series converges to 0 INF | | if the series equals to ∞∞ \infty NIF | | if the series equals to −∞−∞ -\infty [/table] [table] WD | | if the series is not well defined | | [/table] In the second answer field select one of the following reasons that can be used to prove your claim in the first answer field: [table] DT | | The Divergency Test IT | | The Integral Test AS | | The Alternating Series Test RO | | The Root Test RA | | The Ratio Test [/table] [table] D | | The sequence of summands decreases to 00 0 L | | The limit of summands exists and equals to 00 0 C | | Comparison with a geometric series ∑∞𝑛=0𝑞𝑛∑n=0∞qn \sum_{n=0}^{\infty}q^n CH | | Comparison with the harmonic series AH | | Comparison with the alternating harmonic series [/table] [table] P | | Comparison with 𝑝pp-series, where 𝑝>1p>1p > 1 LP | | Comparison with 𝑝pp-series, where 𝑝<1p<1p < 1 [/table] [table] ∑𝑛=1∞sin(𝜋9𝑛)∑n=1∞sin⁡(π9n) \sum_{n=1}^\infty \sin\left( \frac{\pi }{ {9} n } \right) | | because ∑𝑛=1∞cos(𝜋10𝑛)∑n=1∞cos⁡(π10n) \sum\limits_{n=1}^{\infty} \cos\left( \frac{\pi }{ {10} n } \right) | | because [/table]

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MAT137Y1 LEC 20249: Calculus with Proofs (all lecture sections) Pre-Class Quiz 61(13.13 and 13.14)

Let 𝑓 be a continuous function with domain 𝑅 . Assume 𝑓 is POSITIVE and DECREASING. Assume lim 𝑥 → ∞ 𝑓 ( 𝑥 ) = 0 . Which of the following statements MUST be true? Select all the correct answers.

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类似问题

Puzzle for you. Let [math: s] be a convergent series. The series consisting of the positive terms of [math: s] is [math: s+]s^+ and the series consisting of the negative terms is [math: s−]s^-. Which of the following statements is not always true.

Puzzle for you. Let \(s\) be a convergent series. The series consisting of the positive terms of \(s\) is \(s^+\) and the series consisting of the negative terms is \(s^-\). Which of the following statements is not always true.

Which of the following statements is NOT true?

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