Convergence and Divergence Problem: From 12-1, recall that a series converges to f(x)

Chapter 12, Problem 6

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Convergence and Divergence Problem: From 12-1, recall that a series converges to f(x) for a particular value of x if the partial sums of P(x) approach the value of f(x) as the number of terms in the partial sum approaches infinity. In 5, the series P(x) converges to ln x when x = 1.2 and x = 1.95, but the series diverges for x = 3. In this problem you will see why this is true. a. Make a table of values for the first few terms of P(3). What is happening to the absolute value of the terms? b. By appropriate use of lHospitals rule, show that the absolute value of the nth term for P(3) approaches infinity as n approaches infinity. Explain how this fact indicates that the series for P(3) cannot possibly converge. c. Make a table of values for the first few terms of P(1.2). Show that these terms approach zero for a limit as n approaches infinity, and thus the series could converge. d. In 3, you observed that the value of the tail of the series for sin x that remained after the nth partial sum was smaller in absolute value than the absolute value of the first term of the tail. Is this observation true for P(1.2)? Justify your answer.

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