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5 ημέρες πριν · A power series in powers of x - x0 (or just power series) is an infinite series of the form where x0 is a fixed number, x is a variable, and the sequence are often called the coefficients of the series. The power series is said to converge to the function f ( x) if the sequence of partial sums converges to f ( x) at a point x.
3 ημέρες πριν · The approximation ( +) and its equivalent form + ( + ( +)) can be obtained by rearranging Stirling's extended formula and observing a coincidence between the resultant power series and the Taylor series expansion of the hyperbolic sine function.
3 ημέρες πριν · In calculus, Taylor's theorem gives an approximation of a -times differentiable function around a given point by a polynomial of degree , called the -th-order Taylor polynomial. For a smooth function, the Taylor polynomial is the truncation at the order of the Taylor series of the function.
2 ημέρες πριν · In this paper, we provide an overview of fractal calculus as applied to fractal sets. We introduce the concept of power series defined on fractal sets and explore series solutions for second \(\alpha \)-order fractal linear differential equations, particularly focusing on solutions near ordinary points.We also investigate the radius of convergence for these fractal series, elucidating the ...
2 ημέρες πριν · Definition. A polynomial expression is an expression that can be built from constants and symbols called variables or indeterminates by means of addition, multiplication and exponentiation to a non-negative integer power.
2 ημέρες πριν · Power series Theorem 1.1 (The Convergence Theorem for Power Series). If the power series ∞ X n =0 c n (x - a) n converges absolutely at x = b, then it converges absolutely for all x in the interval [a-| a-b |, a + | a-b |], i.e. the closed interval centered at a with radius which is the distance between a and b.
1 ημέρα πριν · A recurrence relation is an equation that uses recursion to relate terms in a sequence or elements in an array. It is a way to define a sequence or array in terms of itself.
