# Mathematical analysis and Series (mathematics)   Analysis is the branch of mathematics dealing with continuous functions, limits, and related theories, such as differentiation, integration, measure, infinite sequences, series, and analytic functions.

- Mathematical analysis

The study of series is a major part of calculus and its generalization, mathematical analysis.

- Series (mathematics) 5 related topics ## Calculus

Mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithmetic operations.

Mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithmetic operations.      These two branches are related to each other by the fundamental theorem of calculus, and they make use of the fundamental notions of convergence of infinite sequences and infinite series to a well-defined limit.

In mathematics education, calculus denotes courses of elementary mathematical analysis, which are mainly devoted to the study of functions and limits. ## Archimedes

Approximation of pi; defining and investigating the spiral that now bears his name; and devising a system using exponentiation for expressing very large numbers.

Approximation of pi; defining and investigating the spiral that now bears his name; and devising a system using exponentiation for expressing very large numbers.              Archimedes anticipated modern calculus and analysis by applying the concept of the infinitely small and the method of exhaustion to derive and rigorously prove a range of geometrical theorems, including: the area of a circle; the surface area and volume of a sphere; area of an ellipse; the area under a parabola; the volume of a segment of a paraboloid of revolution; the volume of a segment of a hyperboloid of revolution; and the area of a spiral.

He expressed the solution to the problem as an infinite geometric series with the common ratio 1⁄4: ## Mathematics                   Mathematics is an area of knowledge that includes such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and the spaces in which they are contained (geometry), and quantities and their changes (calculus and analysis).

He developed formulas for calculating the surface area and volume of solids of revolution and used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, in a manner not too dissimilar from modern calculus. ## Sequence

Enumerated collection of objects in which repetitions are allowed and order matters.

Enumerated collection of objects in which repetitions are allowed and order matters.    In mathematical analysis, a sequence is often denoted by letters in the form of ''.

In particular, sequences are the basis for series, which are important in differential equations and analysis. ## Trigonometric functions

Angle of a right-angled triangle to ratios of two side lengths.

Angle of a right-angled triangle to ratios of two side lengths.         Modern definitions express trigonometric functions as infinite series or as solutions of differential equations.

However, in calculus and mathematical analysis, the trigonometric functions are generally regarded more abstractly as functions of real or complex numbers, rather than angles.