Use known Maclaurin series to find Laurent expansions of the following functions about each of their singular points, stating where each expansion is valid. 2 1 - exp2z (a) (b) (c) (d) (e) 1 - COS z z4 1 z(x − 1) - 1 z3 (2-2) 1 z(2-2)3 (f) sin (g) (1) ez z(22+1)
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- Q3. Use Taylor series ( with a-2 and n-4) to evaluate the following: cosx dx In (1 +x) b) lim a) cos xConsider the expansion y = p=0 96pxop with coefficients given by a6p = (-1)Pao/(2p)!. Which function does this series correspond to? ao exp(x6). Cao sin(x). ao sin(x³). Cao cos(x6). Cao exp(x3). ao cos(x³).Find cosine series expansion for the following function and sketch the graph ,0Asap plzConsider the following two Taylor Series (about x = 0): x3 x7 -1 tan x=x- + + 3 5 7 x3 3x5 5x7 -1 sin x=x+ + + 6 40 112 (a) Find the Taylor Series expansion for f(x) and including the xⓇ term. = -1 (tan¹x) (sin x) about x = 0, up to -1 (b) Use terms up to and including the x¹³ term in the expansion for tan x, together with an appropriate value for x, to find an approximation for π. Write your approximation correct to 4 decimal places.Can you do a frayer model as best of your ability with this equation5. (a)Find the Fourier sine series for f(x) = x³ – n²x on the interval [-7, T]. (b) By plugging in a suitable value for x, find the limit of the convergent series 13 33 53ملوا Find the Fourier sine series for f(x) 1.45 2) con on - L < a < L ة الهنـــدسـةQ5: a\. Let f(x) = x³ - 3x² - 1,x2 2. State and prove the theorem of derivative of inverse functions and use them to find the value of df-1/dx at the point x = -1 = f(3). b\ Write the Pascal's triangle for values of n = 6, then use it to determine the binomial series expansion of (3 - 2x)6, and simplify the result.