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Determine whether the series converges or diverges.$ \displaystyle \sum_{n = 1}^{\infty} \frac {e^n + 1}{ne^n + 1} $

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Diverges

Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 4

The Comparison Tests

Sequences

Series

Oregon State University

Baylor University

University of Michigan - Ann Arbor

University of Nottingham

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.

02:08

Determine whether the seri…

00:49

02:15

03:51

let's determine whether the Siri's convergence or diverges no. So here. Let's use the Lim comparison test from the section in the textbook. So here let me call this an and then let me define B end to be e n over Andy End. And then I could cancel, too. You just get one over Ed, and the reason I wrote beyond this way is because I just look it. Anne and I look at it, the larger terms in the numerator as N goes to infinity and same thing on the denominator. And then here I just cancelled out the easier than power. Now let's look at the limit of an over Bian. Then if I divide by one over and that's the same thing, just multiplying by N. And then here I should keep writing that limit. So I forgot to write the limit. Let me write that in there, Lim Lim. Now, as we take that limit first, let's go in, and you could use low Patel's rule here if you want. But here, let's just divide top and bottom by Andy to the end. So we'LL divide here the whole fraction by this term, and we have one plus one over Ian after canceling one plus one over any end. And then as we take that limit, since either then goes to infinity this denominators going to infinity. So that goes to zero The same thing for that denominator. So we just have one plus zero over one plus zero, which is one we know that the sum of the beings diverge Diverges. This is just a harmonic series. Or you could even use the pizza's here. Pee test with P equals one because that's the power of the end over here on the formula for being, so it diverges for that value of P. On the other hand, we also know that the limit that we just computed satisfies this inequality here because we're just computer that was equal to one in any time. This limit is bigger than zero in less than infinity. You can use the Lim comparison test and this tells you that a sum of a N and bien both converge or bull diverge. And since we already know that being diverges Bye, lct by Lim comparison test some abbreviating this El Sisi. We know that the sum of the A end. So let's write that out formally either the N plus one and either then plus one also beverages, and that's our final answer.

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