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Determine whether the series is convergent or divergent.$ \displaystyle \sum_{n = 1}^{\infty} \frac {\sqrt n + 4}{n^2} $

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Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 3

The Integral Test and Estimates of Sums

Sequences

Series

Campbell University

Oregon State University

Baylor University

Idaho State University

Lectures

01:59

In mathematics, a series is, informally speaking, the sum of the terms of an infinite sequence. The sum of a finite sequence of real numbers is called a finite series. The sum of an infinite sequence of real numbers may or may not have a well-defined sum, and may or may not be equal to the limit of the sequence, if it exists. The study of the sums of infinite sequences is a major area in mathematics known as analysis.

02:28

In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed. Like a set, it contains members (also called elements, or terms). The number of elements (possibly infinite) is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence. Formally, a sequence can be defined as a function whose domain is either the set of the natural numbers (for infinite sequences) or the set of the first "n" natural numbers (for a finite sequence). A sequence can be thought of as a list of elements with a particular order. Sequences are useful in a number of mathematical disciplines for studying functions, spaces, and other mathematical structures using the convergence properties of sequences. In particular, sequences are the basis for series, which are important in differential equations and analysis. Sequences are also of interest in their own right and can be studied as patterns or puzzles, such as in the study of prime numbers.

01:02

Determine whether the seri…

01:04

01:09

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00:49

06:39

to determine the convergence of this Siri's I could be helpful to split it up. So we have the sun for many girls. Want to infinity? Ah, in the squared event plus four over in scored. I'm going to split that up like so? Because if we split up the sun and show that each piece converges, then the overall sum itself will converge. So since we're summing sums, we can split these up. Well, you know, one son of just squared of n over and squared Nowthe squared of end over and squared is scenting his end of the one half over and squared, which is one over and two minds one half which is into the three halves. So we have won over and three halves plus the sun from an equals one to infinity for over and squared. And since this is a four over and squared, sorry could be equal to four times one over and squared. We can split that up as well. So we end up with the son from an equals one to infinity, one over n to the haves plus four times the sun and equals one insanity of when over and squared. And now bye Pee Siri's We have won over end to the P P greater than one. That means that the Siri's converges So the first son converges. The second son also converges Pipi, Siri's and the sum of two conversion. Siri's well itself be convergent, so our overall Siri's is convergent.

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