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Determine whether the series is convergent or divergent. If it is convergent, find its sum.$ \displaystyle \sum_{k = 1}^{\infty} \frac {k^2}{k^2 - 2k + 5} $

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The series divergesHint : Use divergence test

Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 2

Series

Sequences

Campbell University

University of Nottingham

Boston College

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.

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Determine whether the seri…

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Let's go ahead and determine whether the syriza's commercial, our diversion and if it's conversion will go ahead and find that some as well. But that's a big if we'LL see. So one option here would be to try to factor into maybe do partial fractions, perhaps telescoping. But this is what the book uses. But before I go that far that will take a bit of work. I would try the diversions test. This is also weren't this way by the author, So Test four divergence. So to use this test here, this is usually easier, easier test, so should often be done first. So it's a way to use it is you just take the limit as Kay goes to infinity of this term over here. Solis, Right? Since the limit is Kay goes to Infinity case, where K squared minus two K plus five. If you can't see what this answer is, you can go ahead and let's even Let's divide a top and bottom of this bye case where? So we really haven't changed the problem because we divided top and bottom by the same thing. Then we have one over one minus two over K and then plus five over K square. And as you look, eh, go to infinity thes to terms. They're going to go to zero and you just have one over one in the limit. So our limit equals one. Therefore, let me take a step back. Since the limit equals one and this is not equal to zero, we'LL have to say the final. It's over here. The Siri's is diversion. Bye, the test for diversions and that's your final answer.

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