Questions
MAT137Y1 LEC 20249: Calculus with Proofs (all lecture sections) Pre-Class Quiz 61(13.13 and 13.14)
Multiple choice
Let ๐ ย be a continuous function with domain ๐ . Assume ๐ is POSITIVE and DECREASING. Assume lim ๐ฅ โ โ ๐ ( ๐ฅ ) = 0 . Which of the following statements MUST be true? Select all the correct answers.
Options
A.โ
๐
=
1
โ
(
โ
1
)
๐
๐
(
๐
)
ย is divergent.
B.IF
โซ
100
โ
๐
(
๐ฅ
)
๐
๐ฅ
ย is convergent, THEN
โ
๐
=
1
โ
๐
(
๐
)
ย is convergent.
C.โ
๐
=
1
โ
(
โ
1
)
๐
๐
(
๐
)
ย is convergent.
D.IF
โ
๐
=
1
โ
๐
(
๐
)
is convergent, THEN
โซ
1
โ
๐
(
๐ฅ
)
๐
๐ฅ
ย is convergent.
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Step-by-Step Analysis
We are given a continuous, positive, decreasing function f on the real line with limit zero as x โ โ. We will analyze each statement in light of standard convergence tests.
Option 1: โ_{n=1}^โ (-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 โ (-1)^......Login to view full explanationLog in for full answers
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