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Find all values of $ c $ for which the following series converges.$ \displaystyle \sum_{n = 1}^{\infty} \left( \frac {c}{n} - \frac {1}{n + 1} \right) $
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Calculus 2 / BC
Chapter 11
Infinite Sequences and Series
Section 3
The Integral Test and Estimates of Sums
Sequences
Series
Missouri State University
Oregon State University
Harvey Mudd College
Baylor 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.
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we'd like to know the values of sea that allowed the Siri's to converge. So here one option is to go ahead and try the integral test. And other option is they're just proceed out your break. Leah's follows. So here, let me go ahead and just rewrite this C A. C minus one plus one and then I'll split ups in terms here. C minus ones constant so I can pull that out. Notice that this first Siri's over here. This's a telescoping Siri's, and this can be evaluated, and we will actually see that the Siri's converges. So let's just focus on this series for a second. And let's rewrite it as a limit. Kay goes to infinity. Now let's go ahead and evaluate that partial son. Okay? We would keep going all the way until you plugged in less maybe do a few terms before okay was through K minus two and then came on this one and then kay. And now we should cancel is much as we can. So we see the first term we'LL stay there the one But all these intermediate terms cancel out all, even all the way until you get to came on this one. Okay, However, this one over came on. This one will also stay. So you just have the one in the final term. That's day. And when you take the limit, this fraction or this expression goes toe one because the fraction goes to zero. So that shows that this summit is just equal to one so we can rewrite the whole expression as one plus C minus one someone over end. However, if you look at the other Siri's here, our circle is in black. This's harmonic, or you can say it. It's P Siri's with people's one, So this diverges. So unless we want this Siri's to diverge, we should go ahead and take C equals one, because then they'LL wipe off this here, and that will ensure that our some conversions to just the number one so converges. If and only if C equals one
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