- Please Ignore The 1st Question A Proceed With The Following Ones This Exercise Is Very Critical And I Would Encourage 1 (215.1 KiB) Viewed 33 times
Please, ignore the 1st question a proceed with the following ones , this exercise is very critical and I would encourage
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Please, ignore the 1st question a proceed with the following ones , this exercise is very critical and I would encourage
Please, ignore the 1st question a proceed with the following ones , this exercise is very critical and I would encourage to take on only if you are able to complete it all in one go, not part by part..
A fundamental idea in the study of series is that of comparison. We look at and feel that it should behave much like n² Σ; n=1 2 so we try to formalize this idea. We write <=, and then we realize that this inequality goes the wrong way; a<b, and diverges. From this information, one is not allowed to conclude anything about 21 Still, we have the feeling that we should be able to use what we know about . And our Zn=1 feeling is right! It's just that we need a variation on the comparison test. The Limit Comparison Test gives us what we need. Limit Comparison Test: If an and b,, are sequences of positive numbers such that lim, is a positive real number (greater than zero, but finite), then l converges. converges if and only if Σ The goal of this lab is to give you a feel for how to choose a comparison series in order to use the Limit Comparison Test. 2m 1 m² Problem 1. Use the Limit Comparison Test to determine if converges or diverges. Problem 2. Consider the series 1 723/2+18 n=1 and g(x) = (a) Use Desmos to graph the functions f(x)= on the same axes. Set the scale on the horizontal axis so the a-values go from 1 to 100. We are interested in comparing the functions for large values of x, and when z is large, both f(x) and g(x) are quite small. Thus you should set a scale on the vertical axis to view the y-values between 0 and 1. Paste the graph into your work. (b) Use Desmos to plot the function f defined in part a, together with h(x) = on the same axes. Paste the results into your work. You may have to experiment to find a range of y-values that will enable you to see both functions on the same axes. 2 f(x) f(x) (c) Use your computer to calculate lim and lim, for the functions in parts a and b above. (d) Look at the results of parts a through c and the statement of the Limit Comparison Test. Should you compare the series 1/4 with or with ? Explain your choice after considering which term dominates the denominator for large values of n. (e) Does the series -1 + converge? Use the Limit Comparison Test and your knowledge of p-series to justify your conclusion. 1+² Problem 3. Let us consider a different series Experiment with graphs and limits until you find a p-series with the appropriate value of p so you can apply the Limit Comparison Test to this new series. Use the same techniques that you did in answering Problem 2 above. Note your attempts and why you rejected the failures, as well as why you chose the p you did. Does this series converge or diverge? Why? 1+n² Problem 3. Let us consider a different series Σn-1. Experiment with graphs and limits until n³+n2. you find a p-series with the appropriate value of p so you can apply the Limit Comparison Test to this new series. Use the same techniques that you did in answering Problem 2 above. Note your attempts and why you rejected the failures, as well as why you chose the p you did. Does this series converge or diverge? Why? Problem 4. Look next at n+ √n n7/3 + n² n=1 . Again experiment with graphs and limits in order to choose a suitable comparison series. This time consider the dominant term in the numerator as well as the dominant term in the denominator. If you ignore the other terms, what p-series does this series most resemble? Does this series converge or diverge? Problem 5. All of the examples have been series of the form S=²+³. to what p-series would you compare S in order to use the Limit Comparison Test?
A fundamental idea in the study of series is that of comparison. We look at and feel that it should behave much like n² Σ; n=1 2 so we try to formalize this idea. We write <=, and then we realize that this inequality goes the wrong way; a<b, and diverges. From this information, one is not allowed to conclude anything about 21 Still, we have the feeling that we should be able to use what we know about . And our Zn=1 feeling is right! It's just that we need a variation on the comparison test. The Limit Comparison Test gives us what we need. Limit Comparison Test: If an and b,, are sequences of positive numbers such that lim, is a positive real number (greater than zero, but finite), then l converges. converges if and only if Σ The goal of this lab is to give you a feel for how to choose a comparison series in order to use the Limit Comparison Test. 2m 1 m² Problem 1. Use the Limit Comparison Test to determine if converges or diverges. Problem 2. Consider the series 1 723/2+18 n=1 and g(x) = (a) Use Desmos to graph the functions f(x)= on the same axes. Set the scale on the horizontal axis so the a-values go from 1 to 100. We are interested in comparing the functions for large values of x, and when z is large, both f(x) and g(x) are quite small. Thus you should set a scale on the vertical axis to view the y-values between 0 and 1. Paste the graph into your work. (b) Use Desmos to plot the function f defined in part a, together with h(x) = on the same axes. Paste the results into your work. You may have to experiment to find a range of y-values that will enable you to see both functions on the same axes. 2 f(x) f(x) (c) Use your computer to calculate lim and lim, for the functions in parts a and b above. (d) Look at the results of parts a through c and the statement of the Limit Comparison Test. Should you compare the series 1/4 with or with ? Explain your choice after considering which term dominates the denominator for large values of n. (e) Does the series -1 + converge? Use the Limit Comparison Test and your knowledge of p-series to justify your conclusion. 1+² Problem 3. Let us consider a different series Experiment with graphs and limits until you find a p-series with the appropriate value of p so you can apply the Limit Comparison Test to this new series. Use the same techniques that you did in answering Problem 2 above. Note your attempts and why you rejected the failures, as well as why you chose the p you did. Does this series converge or diverge? Why? 1+n² Problem 3. Let us consider a different series Σn-1. Experiment with graphs and limits until n³+n2. you find a p-series with the appropriate value of p so you can apply the Limit Comparison Test to this new series. Use the same techniques that you did in answering Problem 2 above. Note your attempts and why you rejected the failures, as well as why you chose the p you did. Does this series converge or diverge? Why? Problem 4. Look next at n+ √n n7/3 + n² n=1 . Again experiment with graphs and limits in order to choose a suitable comparison series. This time consider the dominant term in the numerator as well as the dominant term in the denominator. If you ignore the other terms, what p-series does this series most resemble? Does this series converge or diverge? Problem 5. All of the examples have been series of the form S=²+³. to what p-series would you compare S in order to use the Limit Comparison Test?
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