By George A. Anastassiou, Iuliana F. Iatan
Real research is a self-discipline of extensive research in lots of associations of upper schooling, since it comprises precious options and basic ends up in the learn of arithmetic and physics, of the technical disciplines and geometry. This booklet is the 1st certainly one of its sort that solves mathematical research issues of all 4 similar major software program Matlab, Mathcad, Mathematica and Maple. in addition to the elemental theoretical notions, the ebook comprises many routines, solved either mathematically and by way of computing device, utilizing: Matlab 7.9, Mathcad 14, Mathematica eight or Maple 15 programming languages. The publication is split into 9 chapters, which illustrate the appliance of the mathematical ideas utilizing the pc. every one bankruptcy provides the basic thoughts and the weather required to unravel the issues contained in that bankruptcy and finishes with a few difficulties left to be solved via the readers. The calculations may be proven through the use of a particular software program similar to Matlab, Mathcad, Mathematica or Maple.
Read or Download Intelligent Routines: Solving Mathematical Analysis with Matlab, Mathcad, Mathematica and Maple PDF
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Additional resources for Intelligent Routines: Solving Mathematical Analysis with Matlab, Mathcad, Mathematica and Maple
Example text
Whence √ 1 n an = < 1 n→∞ e and using the root test it follows that lim n n≥1 1+ 2 1 n n converges. 22 (The ratio test, see [15], p. 94): Let positive series such that an = 0 for any n ≥ 1. Assume that: n≥1 an be a an+1 = λ. n→∞ an lim Then we have the following: A)If λ < 1, then the series n≥1 an is convergent; B) If λ > 1, then the series n≥1 an is divergent; C)If λ = 1, then the series n≥1 an may be convergent or it may be divergent, namely we do not have a definite conclusion. 23. ) (2n)! n tan n≥1 π 2n+1 n · an , a > 0.
N! 9 (see [15], p. 282). In order to evaluate the remainder, one can employ the formula: Tn (x) = f (a) + (x − a) f (a) + n+1 Rn (x) = (x − a) f (n+1) (a + θ (x − a)) , 0 < θ < 1. (n + 1)! 10 (see [15], p. 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} .
Lim n n→∞ x→0 ln n+1 x lim We shall compute lim n·ln n→∞ n n+1 n = lim ln n + 1 n→∞ n = ln lim n→∞ 1 1 + n1 n = ln e−1 = −1. We have used the fact that lim x→∞ 1+ 1 x x = e. Finally, one obtains lim n n→∞ an −1 an+1 = ln 7 · ln e−1 = − ln 7 < 1 and using the Raabe’s and Duhamel’s test it follows that 7ln n n≥1 diverges. 4 Tests of Convergence and Divergence of Positive Series 29 or in Maple 15: c) We shall have n an −1 an+1 n! α (α + 1) (α + 2) · · · · · (α + n − 1) α (α + 1) (α + 2) · · · · · (α + n − 1) (α + n) −1 · (n + 1)!