4. Taylor Series Expansion of a function: (i) Find the first four terms of the Taylor Series of f(x) = cos(x) about the point x = x_0. (ii) Show that the general expression for the Taylor expansion for f(x) = cos(x) about the point x_0 = ??/2 can be written: cos(x) = T(f, ??/2)(x) = ??_{l=0}^{?} (-1)^{l+1} rac{(x - ??/2)^{2l+1}}{(2l + 1)!} (iii) Use the first two terms in the above expansion to compute the value of cos(0.55??). (iv) Recall the Maclaurin expansion of f(x) = cos(x) can be written: cos(x) = T(f, 0)(x) = ??_{m=0}^{?} (-1)^m rac{x^{2m}}{(2m)!} Use the first two terms in this expansion to compute cos(0.55??). (v) Use your calculator to compute the true answer for cos(0.55??) and state whether your approximate Taylor or Maclaurin series expansion is more accurate.
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The Taylor series expansion for sine, cosine and exponential functions are given by: sin(x) = x - (x^3)/3! + (x^5)/5! - (x^7)/7! + ... and cos(x) = 1 - (x^2)/2! + (x^4)/4! - (x^6)/6! + ... and e^x = 1 + x + (x^2)/2! + (x^3)/3! + ... Use these and substitution into the exponential series to verify the famous Euler's equation: e^(iθ) = cos(θ) + i sin(θ) where i = √-1 and θ is a real number. Choose a specific value of θ to obtain another of Euler's results: e^(iπ) + 1 = 0 Bonus: Use (1) (do not expand) to prove DeMoivre's Theorem: (cos(θ) + i sin(θ))^n = cos(nθ) + i sin(nθ).
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(a) Find the Taylor series for f(x) = ln x centered at x = 1 [3 marks] (b) Find the Maclaurin series for f(x) = sin x [3 marks] (c) Find the nth term of the series 1/2 + 2/3 + 3/4 + 4/5 + ... Using the nth term test, investigate the convergence of the series above. [4 marks] (d) Investigate the series 1/∑1 + 1/∑2 + 1/∑3 + 1/∑4 + ... for convergence or divergence using the comparison test. [2 marks] (e) Find the range of values of x for convergence of the series 1/(1 2) + x/(2 3) + x^2/(3 4) + x^3/(4 5) + ... + x^(n-1)/(n(n+1)) [Hint: use Ratio Test] [4 marks] (f) Find the odd and even parts of the function, f(x) = (1-x)/(1+x). [4 marks] (g) Given f(x) = 2x^2 - 7x + 10 on ]2,5[, find all values of c in the interval. [3 marks] (h) Express Z = -4 + 3i in polar form. [3 marks] (i) Find the modulus and the argument of the complex number Z = ∑5 - 2i. [4 marks]
Find the Taylor series for $f(x)$ centered at the given value of $a$ . [Assume that $f$ has a power series expansion. Do not show that $R_{n}(x) \rightarrow 0.1]$ $$f(x)=\cos x, \quad a=\pi$$
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Taylor and Maclaurin Series
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