# Trigonometric Series, Volume I & II Combined

According to 14 , we have. Now we prove the right-hand side inequality In this example we propose the following improvement of It suffices to show that the following mixed logarithmic-trigonometric-polynomial function [ 11 ]. Given that. As a result of this selection, the Natural Approach algorithm yields the polynomial. B i are the Bernoulli numbers; see, e. Let us consider the real analytical function.

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Also, the restriction. A similar analysis shows us that only the following refinements of inequality 20 are possible. The results of our analysis could be implemented by means of an automated proof assistant [ 31 ], so our work is a contribution to the library of automatic support tools [ 32 ] for proving various analytic inequalities.

## Trigonometric Series, Third Edition, Volume I & II Combined (Cambridge Mathematical Library)

Our general algorithm associated with the natural approach method can be successfully applied to prove a wide category of classical MTP inequalities. For example, the Natural Approach algorithm has recently been used to prove several open problems that involve MTP inequalities see, e. It is our contention that the Natural Approach algorithm can be used to introduce and solve other new similar results.

The power series of the function cos 2 n x is an alternating sign series. Therefore, for the above power Taylor series, it is not hard to determine depending on m which partial sums i. Assuming the following representation of the function cos 2 n x in power Taylor series. Such estimation of the function cos 2 n x and the use of corresponding Taylor approximations will be the object of future research. Competing interests. Authors would like to state that they do not have any competing interest in subject of this research. All authors participated in every phase of research conducted for this paper.

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National Center for Biotechnology Information , U. Journal of Inequalities and Applications. J Inequal Appl. Published online May Author information Article notes Copyright and License information Disclaimer. Corresponding author. Received Feb 28; Accepted May 2. This article has been cited by other articles in PMC.

Trigonometric Ratio & Identities - summation of Series in Trigonometry - PART-26 - Class-11 CBSE/JEE

Keywords: mixed trigonometric-polynomial functions, Taylor series, approximations, inequalities, algorithms, automated theorem proving. Introduction and motivation In this paper, we propose a general computational method for reducing some inequalities involving trigonometric functions to the corresponding polynomial inequalities. The natural approach method and the associated algorithm The following two lemmas [ 8 ] related to the Taylor polynomials associated with sine and cosine functions will be of great help in our study.

Based on the above results, we have the following.

## Trigonometric Series by Zygmund - AbeBooks

Open in a separate window. Theorem 8 The Natural Approach algorithm is correct. Proof We first prove the left-hand side inequality Conclusions and future work The results of our analysis could be implemented by means of an automated proof assistant [ 31 ], so our work is a contribution to the library of automatic support tools [ 32 ] for proving various analytic inequalities. Contributor Information Tatjana Lutovac, Email: sr. Following the convention used by the Wolfram Language , the inverse trigonometric functions defined in this work have the following ranges for domains on the real line , illustrated above.

Complex inverse identities in terms of natural logarithms include. Abramowitz, M. New York: Dover, pp. Apostol, T. Waltham, MA: Blaisdell, pp.