Yоu MUST NOT use аnything during this exаm except а nоtes page as explained under Final Exam Discussiоn.Read the following statement: ---- I am familiar with the requirements of the Mason Honor Code.I have complied with those requirements in all respects in relation to this exam.I hereby declare that the following work is entirely my own and has been completed in accordance with the directions of the Instructor and exam supervisor(s).----Enter your full name `full name` and your Mason G Number `Mason G Number` to make the above declaration.(No points, but your exam will not be graded if you do not complete this question.)
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Infinite Prоcesses аnd Alternаtive RepresentаtiоnsWrite explanatiоns that addresses each of the prompts below. Show your ideas, with reasons. If uncertain about something, write what you DO know and articulate what you are unsure about. (this is worth points).Part (a): Alternating Series Approximation Test (7 points)Consider an alternating series ∑n=1∞(-1)n+1an where an>0 for all n≥1.(i) State the conditions under which an alternating series converges. (2 points)(ii) If an alternating series ∑n=1∞(-1)n+1an converges to a sum S, explain how the partial sum Sn=∑k=1n(-1)k+1ak can be used to approximate S. (2 points)(iii) For a convergent alternating series, explain how to find an upper bound for the error when approximating the sum S using the partial sum Sn. (3 points)Part (b): Power Series and Approximations (7 points)(i) Write the Maclaurin series for sin(x). (2 points)(ii) Using the Maclaurin series for sin(x), find the Maclaurin series for sinx2. (2 points)(iii) Show that the series for sinx2 is an alternating series when x2>0, and verify that it satisfies the conditions for the alternating series test. (2 points)(iv) Using the alternating series approximation, determine how many terms of the series for sin0.52=sin0.25 you would need to include to ensure that the error in the approximation is less than 0.01. Calculate this approximation. (2 points)Evaluation criterion:Correct mathematical definitions and notationsClear explanations of conceptsLogical organization and flow of ideasPrecise mathematical language and terminology