6. By the Alternating Series Test, show that the following series expansion converges regardless of x, as long as x is finite. Use the growth rates of sequences {Theorem 10.5) to help with your evaluation. C0 . . x3 x5 (_1)k-1xak-1 slnx = x+ = I l _ f 3. 5. H (2!: 1). A common approximation in physics when small angles are considered is to truncate the sine series above at k = 1, that is, sin x = x. For three different angles: 4, 22\7. Do the following series (choose one:) converge absolutely / converge conditionally / diverge / can't tell by test according to the Ratio Test? To show the reason for your answer, evaluate the ratio at the limit. K = 1 (-1)* 6k (5k - 6*)8. Do the following series {choose one:) converge I diverge {can't tell by test according to the Root Test? To show the reason for your answer, evaluate the root at the limit. its)\" k=1 iris) P? H ,.i. 9. Approximate the function f(x) = sin(x), centered at 2n/3. Use a cubic polynomial. What is the value of this polynomial when x = 13n/20? Show the numerical value for each term. Extend the cubic polynomial to a power series, writing it as a summation of an infinite number of terms. Make the summation index k, write 2n/3 for a and f(*)(sin 2n/3) for the derivatives d* (sin a)/dx*. Had you tried to find the k value needed in order to obtain |S - Sk| S ak+1 $ 0.0001 for the angle 2n/3 in Problem 6, you would have found k = 5. How many terms of the series constructed for this problem are needed in order to obtain the same accuracy, that is, of better than 0.0001