INFINITE SERIES ANALYSIS IN THE COMPLEX PLANE

Abstract

Background: In this article we want to show when Taylor series function At all points of the circle centered at where is converged inside the circle Methods: To this end, we must show that such is large enough to zero, but in surveys that we It is easily seen that each subsequent term in the expansion for is much smaller than the previous one as . However, if a particular value of x , is considered, then the modulus of the n th term in first decreases with increase of the number of the term n but then, after reaching a minimum, starts to increase. Results: This means that the sum in represents a divergent series, and therefore increase of number of terms in will not necessary lead to improvement of the accuracy of approximation. In fact, it will eventually result in deterioration of approximation. At the same time, with a finite number of terms expansion gives perfectly accurate representation of . In view of the difference between and its asymptotic expansion tends to zero as faster than the last term kept in the expansion. Conclusion: To solve this problem we have used of the steepest descent method.