(2 marks) In this question you may find usefull one of the following Maclaurin expansions ex=k=0∑∞​k!xk​,sinx=k=0∑∞​(−1)

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2 Marks In This Question You May Find Usefull One Of The Following Maclaurin Expansions Ex K 0 K Xk Sinx K 0 1 1 (21.97 KiB) Viewed 33 times

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(2 marks) In this question you may find usefull one of the following Maclaurin expansions ex=k=0∑∞​k!xk​,sinx=k=0∑∞​(−1)k(2k+1)!x2k+1​,cosx=k=0∑∞​(−1)k(2k)!x2k​ valid for all x∈R, 1−x1​=k=0∑∞​xk valid for x∈(−1,1) Suppose that the Taylor series for excos(4x) about 0 is a0​+a1​x+a2​x2+⋯+a6​x6+⋯ Enter the exact values of a0​ and a6​ in the boxes below.
Suppose that a function f has derivatives of all orders at a. Then the series k=0∑∞​k!f(k)(a)​(x−a)k is called the Taylor series for f about a, where f(n) is the nth order derivative of f. Suppose that the Taylor series for e2xcos(2x) about 0 is a0​+a1​x+a2​x2+⋯+a4​x4+⋯ Enter the exact values of a0​ and a4​ in the boxes below.