Is summation and integration same?

Summation- Sum of a small numbers of large quantities. Integration- Sum of a large numbers of small quantities. The Summation is a discrete sum whereas Integration is a continuous sum . Here dx is an infinitesimal so that the integral summation is continuous.

Can summations be interchanged?

We know that we can interchange the order of summations here.

What is the relation between integration and summation?

Integration is basically the area bounded by the curve of the function, the axis and upper and lower limits. This area can be given as the sum of much smaller areas included in the bounded area. Summation involves the discrete values with the upper and lower bounds, whereas the integration involves continuous values.

When can limit be interchanged?

lim n → ∞ f n ( 0 ) = lim n → ∞ 1 = 1. if if. The function is not continuous at 0. If we, however, require the convergence to be uniform, the limits can be interchanged.

Does the order of summation matter?

We also know, however, that when a single series is absolutely convergent, the order of summation does not matter. It turns out that the same is true with double series.

Can you interchange derivative and summation?

Interchanging summation and differentiation is possible if the derivatives of the summands uniformly converge to 0, and the original sum converges. This follows from the equivalent criterion for interchanging limits and differentials.

When can you interchange sum and derivative?

Can a limit of an integral be moved inside the integral?

Taking the limit inside the integral is not always allowed. There are several theorems that allow us to do so. The major ones being Lebesgue dominated convergence and monotone convergence theorems. The uniform convergence theorem is a special case of dominated convergence theorem.

How are differentiation and integration inverse processes?

Integration is the way of inverse process of differentiation. Instead of differentiating a function, we are given the derivative of a function and asked to find its primitive, i.e., the original function. These forms worked with different basic functions as we said are the ideas of an integral.

Which came first integration or differentiation?

Actually integration came first. Mathematicians tried to give a method to check area of a given curve. while in trying it they got an idea that there should be definitely opposite process which is slope of a tangent to a given curve.so they invented differentiation for calculating slope of a tangent to given curve.