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MAT137Y1 LEC 20249: Calculus with Proofs (all lecture sections) Pre-Class Quiz 57 (13.1-13.4)
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Use the definition of the series to determine if the following series are convergent or divergent. 1) โ ๐ = 1 โ ๐ ๐ ๐ ( ๐ ๐ ) is [ Select ] divergent convergent . 2) โ ๐ = 1 โ 1 2 ๐ is [ Select ] convergent divergent . 3) โ ๐ = 1 โ ln โก ( ๐ ๐ + 1 ) ย is [ Select ] divergent convergent . Hint: you may need to use the property of logarithmic function to rewrite ln โก ( ๐ ๐ + 1 ) . 4) โ ๐ = 1 โ 1 ๐ 2 + 3 ๐ + 2 is [ Select ] divergent convergent . Hint: you may need to use the partial fraction here to find your partial sum.ย
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To analyze each series, Iโll consider the definition or a standard test that applies to the general form.
Option 1: The series โ_{n=1}^โ cos(nฯ). Since cos(nฯ) = (-1)^n, the terms do not tend to 0 (they alternate between -1 and 1). A n......Login to view full explanationLog in for full answers
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Question text(1) Here is a convergent infinite series 1+1/2+1/4+1/8+1/16+...What kind of infinite series are we dealing with?Answer 1 Question 16[select: , arithmetic, geometric, p-series, harmonic, telescoping, none of the above]What is the fifth partial sum of this series (written in lowest terms)? Answer 2 Question 16[input] What's its sum? Answer 3 Question 16[input] (2). Here is another convergent infinite series 1+1/4+1/9+1/16+1/25+... What kind of infinite series are we dealing with?Answer 4 Question 16[select: , arithmetic, geometric, p-series, harmonic, telescoping, none of the above]What is the third partial sum of this series? Answer 5 Question 16[input] What is the integer part of its sum? Answer 6 Question 16[input] (3) Here is yet another converging infinite series What kind of infinite series are we dealing with?Answer 7 Question 16[select: , arithmetic, geometric, p-series, harmonic, telescoping, none of the above]What is the sum of the first three terms of this series? Answer 8 Question 16[input] What's its sum? Answer 9 Question 16[input] Check Question 16
Consider the series โ ๐ = 1 โ 0.01 . The terms are ๐ ๐ = 0.01 . ย a) Find the following partial sums: ๐ 1 = [ Select ] 0.01 0.02 1 0 0.03 ๐ 2 = ย [ Select ] 0.02 0.04 2 1 0.01 ๐ 3 = [ Select ] 3 4 0.03 0 0.01 ๐ 4 = [ Select ] 0.4 0.04 4 1 0.01 ย b) Find the limits: ย ย lim ๐ โ โ ๐ ๐ = ย [ Select ] infinity 0.04 0.01 negative infinity 0 and lim ๐ โ โ ๐ ๐ = ย [ Select ] negative infinity infinity 0.01 1 0 ย c) Does the series โ ๐ = 1 โ 0.01 ย converge or diverge? [ Select ] The series converges There is not enough information to tell The series diverges ย d) Suppose another series โ ๐ = 1 โ ๐ ๐ ย has some unknown terms ๐ ๐ ย but we know that lim ๐ โ โ ๐ ๐ = 0.01 ย (this means the numbers ๐ ๐ are close to 0.01, but not necessarily equal to 0.01.)ย What can be said about the convergence of the series โ ๐ = 1 โ ๐ ๐ ย ? [ Select ] The series converges The series diverges There is not enough information to tell
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