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Algebra Problem No.8

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@ Mr. Eric | Saturday, Aug 21, 2021 | 2 minute read | Update at Saturday, Aug 21, 2021

Complete each problem below by hand. Show your work and provide a reason for each step. Calculators are not allowed on this assignment.

  1. The sum of the first 10 terms of a geometric sequence is 4−2−8. If the common ratio is 12, find the first term of the sequence. Use this to determine the 11th term of the sequence.

Answer Answer

  1. It can be difficult to imagein that the sum of an infinite nmber of values is equal to a finite number. When ∑∞n=1an=s, where s is a real number, we say the sum converges, when the sum does not converge we say it diverges. The series ∑∞n=11n is called the harmonic series. Does this series converage or diverge? Explain your reasoing.

Answer

  1. We wish to find the sum of all positive intergers, that is 1+2+3+4+⋯. Note we are not looking for a finite sum, but the infinite sum of all intergers.

HonorsTopic:Convergence

We will try to find a solution using two different methods.

a. Using the sums in the worksheet first determine ∑∞n=1n by calulating ∑∞n=1n−∑∞n=1(−1)n+1n.

Answer

b. Now we will take a different approach. Excluding 1, add the sum in groups of threes until you have enough to see a pattern.

1+(2+3+4)+(5+6+7)+⋯.

Use this pattern to find ∑∞n=1n

Answer

c. Based on your results, does the sum converge or diciverge? Is this what you would expect? Why or why not? Compare and contrast your findings in parts a and b. What can we conclude?

Answer

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