Shów your work. Question 1 [. 小 a) [ : Determine whether the series 1 Σ n=1 (4n- 1)4 is convergent or divergent. b) [. 1: Determine whether the series 5"-1 Σ 3 + 6"-1 n=1
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- 6. Consider xn+1 = (1/3)(2xn - 9/xn2). Does it converge for any nonzero initial point? If so, to what values?Suppose that Newton’s method is applied to find the solution p = 0 ofthe equation e^x − 1 − x −1/2x^2 = 0. It is known that, starting with any p0 > 0, the sequence {pn} produced by the Newton’s method is monotonically decreasing (i.e., p0 > p1 > p2 > · · ·) and converges to 0.Prove that {pn} converges to 0 linearly with rate 2/3. (hint: use L’Hospital rule repeatedly. )determine the radius and convergence of E n=1 5xn/3n2
- 6). Evaluate ∞ n = 0 (−1)n 2n + 1 as 1 0 f(t) dt where f(x) = ∞ n = 0 (−1)nx2n = 1 1 + x2 by identifying it as the value of a derivative or integral of geometric series. please show step by step clearly .Find the series convergent or diverges. If it is convergent find the limit. an = (2 + n3 ) / ( 1 + 2n3 ) a) diverges b) 1/2 c) 0 d) 1/4 e) 3/4Represent the function 2/(1−10x) as a power series f(x)=∑n=0∞ cnxn C0= C1= C2= C3= C4= Find the radius of convergence R=
- For which a does 1/n(ln(n))^a converges? Justify your answer.(Hint In part d) using the Fourier series of this problem together with simple algebra, Note that we are shortly going to prove the fact that if ∑|f^(n)|<∞∑|f^(n)|<∞ then the Fourier series of ff converges uniformly to ff. You may use this fact in your solution.)Show that if $a_{n}>0$ and $\Sigma a_{n}$ is convergent, then $\Sigma \ln \left(1+a_{n}\right)$ is comvergent.
- 1. Sketch f(x) = x^3-5.00x^2+1.01x+1.88, showing the roots near+-1 and 5. Write: x = g(x) = (5.00x^2-1.01x-1.88) (x^2) Find the root starting from x0 = 5,4,1,-1. Explain the results. Find a form x = g(x) of f(x) = 0 in problem 1 that yields convergence to the root near x=1.for what R is sigma n=1 (n!)^2/(K_n)! convergent?Find the limit of the series Σ[(i+1)/4i] where i goes from 0 to infinity.