Use the definition of the series to determine if the following series are convergent or divergent. 1) ∑ 𝑛 = 1 ∞ 𝑐 𝑜 𝑠 ( 𝑛 𝜋 ) is [ Select ] divergent convergent . 2) ∑ 𝑛 = 1 ∞ 1 2 𝑛 is [ Select ] convergent divergent . 3) ∑ 𝑘 = 1 ∞ ln ⁡ ( 𝑘 𝑘 + 1 ) is [ Select ] divergent convergent . Hint: you may need to use the property of logarithmic function to rewrite ln ⁡ ( 𝑘 𝑘 + 1 ) . 4) ∑ 𝑛 = 1 ∞ 1 𝑛 2 + 3 𝑛 + 2 is [ Select ] divergent convergent . Hint: you may need to use the partial fraction here to find your partial sum. Multiple dropdown selections

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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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