An arithmetic series is a sequence of numbers where each term is obtained by adding a constant value to the previous term. For example, the series 2, 5, 8, 11, ... is an arithmetic series with a common difference of 3.
The sum of an arithmetic series can be found using the formula Sn = n/2(a1 + an), where Sn is the sum of the series, n is the number of terms, a1 is the first term, and an is the last term. Alternatively, you can use the formula Sn = n/2(2a1 + (n-1)d), where d is the common difference.
An integral is a mathematical concept used to find the area under a curve or the accumulation of a given function. It is also known as anti-derivative, as it is the inverse operation of differentiation.
Inequalities are used to determine the upper and lower bounds of a given arithmetic series or integral. This is especially useful when trying to find the maximum or minimum value of a function or when proving convergence or divergence of a series.
Arithmetic series and integrals are used in various fields such as physics, economics, and engineering. For example, they can be used to calculate the distance traveled by a moving object, the total revenue of a company, or the amount of material needed for construction projects.