Download e-book for kindle: Intelligent Routines: Solving Mathematical Analysis with by George A. Anastassiou, Iuliana F. Iatan

By George A. Anastassiou, Iuliana F. Iatan

Actual research is a self-discipline of extensive examine in lots of associations of upper schooling, since it includes invaluable options and primary leads to the examine of arithmetic and physics, of the technical disciplines and geometry. This ebook is the 1st one in all its variety that solves mathematical research issues of all 4 similar major software program Matlab, Mathcad, Mathematica and Maple. along with the basic theoretical notions, the ebook comprises many workouts, solved either mathematically and through desktop, utilizing: Matlab 7.9, Mathcad 14, Mathematica eight or Maple 15 programming languages. The e-book is split into 9 chapters, which illustrate the applying of the mathematical techniques utilizing the pc. each one bankruptcy presents the primary recommendations and the weather required to resolve the issues contained in that bankruptcy and finishes with a few difficulties left to be solved by means of the readers. The calculations may be established through the use of  a particular software program corresponding to Matlab, Mathcad, Mathematica or Maple.

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Additional info for Intelligent Routines: Solving Mathematical Analysis with Matlab, Mathcad, Mathematica and Maple

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D) n≥1 Solutions. a) One obtains lim n→∞ an+1 2 · 7 · 12 · · · · · (5n + 2) 5 · 9 · 13 · · · · · (4n + 1) · = lim n→∞ an 5 · 9 · 13 · · · · · (4n + 5) 2 · 7 · 12 · · · · · (5n − 3) 5 5n + 2 = > 1; = lim n→∞ 4n + 5 4 therefore, using the ratio test it results that the series n≥1 2 · 7 · 12 · · · · · (5n − 3) 5 · 9 · 13 · · · · · (4n + 1) diverges. ]2 (2n)! )2 · (n + 1)2 · (2n)! an+1 · = = 2 an (2n + 2)! )2 (2n + 2) (2n + 1) · (2n)! ) (2n)! is convergent. 4 Tests of Convergence and Divergence of Positive Series 25 c) We can write: π π π π π cos 2n+1 an+1 n + 1 tan 2n+2 n + 1 sin 2n+2 n+2 n+1 · · = = · 2 π · 2 π · ; π π π an n tan 2n+1 n cos 2n+2 sin 2n+1 2n+2 2n+1 therefore an+1 1 = n→∞ an 2 lim and using the ratio test one establishes that n tan n≥1 π 2n+1 is convergent.

Computer solution. 9: >> syms n >> a=@(n)tan(1/(n*sqrt(n))); >> limit(a(n),inf ) ans = 1 tan √ n n 35 36 1 Sequences and Series of Numbers 0 or Mathcad 14: or Mathematica 8: or Maple 15: and adding the fact that 1 1 √ > √ n n (n + 1) n + 1 and the tangent function is increasing on − π2 , π2 it results that 1 1 √ tan √ > tan n n (n + 1) n + 1 namely (an )n is monotone decreasing; hence using the Leibnitz’s test it follows that the alternating series is convergent. 6 Problems 7. Test the convergence of the positive series: √ √ n− n−1 .

282). The power series of the form ∞ f (n) (0) n x2 xn (n) x = f (0) + xf (0) + f (0) + . . + f (0) + · · · n! 2! n! 10) namely the particular case of the Taylor series for a = 0 is called the Mac Laurin series. 11. Expand the function f (x) in a series of powers of x: f (x) = 1 1+x ln , x ∈ (−1, 1) 2 1−x b) f (x) = cos3 x, x ∈ R 3x − 5 , x ∈ R\ {1, 3} . c) f (x) = 2 x − 4x − 3 a) f (x) = Solutions. a) One can notice that ⎤ ⎡ f (x) = 1⎢ ⎥ 1 ⎣ln (1 + x) − ln (1 − x)⎦ = [f1 (x) − f2 (x)] , x ∈ (−1, 1) 2 2 f1 (x) f2 (x) and ⎧ ⎧ 1 1 f1 (x) = 1+x f2 (x) = − 1−x ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ f1 (x) = − 1 2 ⎨ f2 (x) = − 1 2 (1+x) (1−x) , 2 2 f1 (x) = (1+x)3 f2 (x) = − (1−x) 3 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ (4) ⎩ (4) 6 6 f1 (x) = − (1+x) f (x) = − .

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