Questions
MAT136H5 S 2025 - All Sections 5.2 preparation check
Numerical
The series converges to the number . (This is given information. You do not need to prove this, but we may be able to prove it before the end of the course.) Compute by adding the first several terms of the series. Notice that for each additional term you add, the sum is closer and closer to . What is the smallest such that gives an approximation of the number that is correct to 3 decimal places? Side note: Some calculators do computations involving and many other transcendental numbers by using sums of series such as the one above (although recently there are also faster methods that calculators can use).
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Step-by-Step Analysis
The problem states that the infinite series converges to a certain number L (though L is not specified here), and asks for the smallest integer n such that the partial sum S_n approximates L to three decimal places.
First, recall what it means to be correct to three decimal places: the approximation must be within 0.0005 of the true value L. That is, we require |S_n − L| < 0.0005.
Next, we ......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 ∑ 𝑛 = 1 ∞ 0.01 . The terms are 𝑎 𝑛 = 0.01 . a) Find the following partial sums: 𝑆 1 = [ Select ] 0.01 0.02 1 0 0.03 𝑆 2 = [ Select ] 0.02 0.04 2 1 0.01 𝑆 3 = [ Select ] 3 4 0.03 0 0.01 𝑆 4 = [ Select ] 0.4 0.04 4 1 0.01 b) Find the limits: lim 𝑘 → ∞ 𝑆 𝑘 = [ Select ] infinity 0.04 0.01 negative infinity 0 and lim 𝑛 → ∞ 𝑎 𝑛 = [ Select ] negative infinity infinity 0.01 1 0 c) Does the series ∑ 𝑛 = 1 ∞ 0.01 converge or diverge? [ Select ] The series converges There is not enough information to tell The series diverges d) Suppose another series ∑ 𝑛 = 1 ∞ 𝑏 𝑛 has some unknown terms 𝑏 𝑛 but we know that lim 𝑛 → ∞ 𝑏 𝑛 = 0.01 (this means the numbers 𝑏 𝑛 are close to 0.01, but not necessarily equal to 0.01.) What can be said about the convergence of the series ∑ 𝑛 = 1 ∞ 𝑏 𝑛 ? [ Select ] The series converges The series diverges There is not enough information to tell
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