By César Pérez López (auth.)

MATLAB is a high-level language and atmosphere for numerical computation, visualization, and programming. utilizing MATLAB, you could research facts, improve algorithms, and create versions and purposes. The language, instruments, and integrated math capabilities show you how to discover a number of methods and achieve an answer swifter than with spreadsheets or conventional programming languages, similar to C/C++ or Java.

MATLAB Differential and necessary Calculus introduces you to the MATLAB language with useful hands-on directions and effects, permitting you to quick in achieving your ambitions. as well as giving a quick creation to the MATLAB atmosphere and MATLAB programming, this booklet presents all of the fabric had to paintings very easily in differential and necessary calculus in a single and a number of other variables. between different center subject matters of calculus, you'll use MATLAB to enquire convergence, locate limits of sequences and sequence and, for the aim of exploring continuity, limits of capabilities. different types of neighborhood approximations of services are brought, together with Taylor and Laurent sequence. Symbolic and numerical concepts of differentiation and integration are lined with various examples, together with functions to discovering maxima and minima, parts, arc lengths, floor components and volumes. additionally, you will see how MATLAB can be utilized to resolve difficulties in vector calculus and the way to unravel differential and distinction equations.

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5 Formal Power Series MATLAB implements the package powseries, which can be loaded into memory using the command with(powseries). This package provides commands which allow you to create, manipulate and perform symbolic calculations with formal power series. The command maple must first be used. e. changes its sign) multconst (pserie, expression): Multiplies each coefficient of the specified power series by the given expression multiply (ps1, ps2): Gives the product of the two specified power series quotient (ps1, ps2): Gives the quotient of the two specified power series powsqrt (pserie): Gives the square root of the specified power series powexp (pserie): Gives the exponential of the specified power series powlog (pserie): Gives the natural logarithm of the specified power series powsin (pserie): Gives the sine of the specified power series powcos (pserie): Gives the cosine of the specified power series powdiff (pserie): Gives the derivative of the specified power series powint (pserie): Gives the indefinite integral of the specified power series 58 Chapter 3 ■ Numerical Series and Power Series compose (ps1, ps2): Composes the specified power series reversion (ps1, ps2): Calculates the reversion of the power series ps1 with respect to the power series ps2 (reversion is the inverse of composition) inverse (pserie): Calculates the multiplicative inverse of the given series evalpow (exprseries): Returns the power series obtained by evaluating the expression exprseries, where the latter is any arithmetic expression involving formal power series, polynomials, or functions that is accepted by the power series package tpsform (pserie, variable, n): Transforms the formal power series pserie into a series of order n in the given variable tpsform (pserie, variable): Transforms the formal series pserie into a series in the given variable with order defined by the global variable Order op (pserie): Returns the internal structure of the formal power series pserie and gives access to its operands op (op (pserie) 4): Returns a table that defines all the coefficients of the formal power series pserie.

Convert (s, polynom): Converts the series s to a polynomial, eliminating the order term convert (s, ratpoly): Converts the series s to a rational polynomial expression. If s is a Taylor or Laurent series, then the rational polynomial expression is the Padé approximation. If s is a Chebyshev series, then the rational polynomial expression is the Chebishev-Padé approximation. convert(s,ratpoly,m,n): Converts the series s to a rational polynomial where the numerator polynomial has degree m and the denominator polynomial has degree n op (s): Shows the internal structure of the series s and allows access to its operands.

Inf) EXERCISE 3-3 Study the convergence and, if possible, find the sum of the following series: ¥ å7 n =1 3 + 2n , n (1 + n) n ¥ n=1 ¥ n åp n , å n =1 ( n + p )! p! We apply the ratio test to the first series: >> maple('a:=n -> (2*n+3)/(n*(n+1)*(7^n)): limit(a(n+1)/a(n), n=infinity)') ans = 1/7 As the limit is less than 1, the series is convergent. We will calculate its sum. 3833972806909634 48 Chapter 3 ■ Numerical Series and Power Series Now we apply the ratio test to the second series: >> maple('a:=n -> n/p^n'); >> maple('limit(a(n+1)/a(n), n=infinity)') ans = 1/p Thus, if p > 1, the series converges, and if p < 1, the series diverges.

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