Sums of convergent series for any desired number terms, which may be infinite, are estimated very accurately by establishing definite rational polynomials. For infinite terms the sum is obtained taking asymptotic limit polynomial. A function with second-degree polynomials both in numerator and denominator found to produce excellent results. different characteristics such as alternating signs co...

In discussing finite difference methods for the solution of hyperbolic par t ia l differential equations, STETrER [1] used est imates on some absolutely convergent Fourier series to prove stabi l i ty and instabi l i ty with respect to uniform convergence. I f / , a complex valued function on the circle, has an absolutely convergent Fourier series, then the n-th power of ] also has an absolutel...

One of the surprising results in an elementary calculus course is that a rearrangement of a conditionaly convergent series may change its sum, even its very convergence. Observing typical textbook examples of this phenomenon, it turns out that during the rearrangement some of the terms are moved arbitrarily large distances from their original locations. Is this necessary? The answer is positive...

is it true that \f(s)\ s^k>0 for cr^O implies that (/(s))-1 is also of the form (1)? In this note, an affirmative answer is supplied.3 Let P be the semigroup of positive integers under multiplication, and let h(P) be the class of all complex functions a on P, a= {an}»~i, for which ||a|| = y^°-i \an\ is finite. We obtain a commutative Banach algebra by defining (aa)n = aan for complex a, (a+b)n ...