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Prove part (i) of Theorem 8.
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Calculus 2 / BC
Infinite Sequences and Series
Oregon State University
Harvey Mudd College
Idaho State University
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.
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.
Prove that part (ii) of Th…
Let's prove par one of theory me in this section and this says If the sum of am converges, then thes to terms are equal here, we're assuming, of course, see is just a constant. So fix real number. So So show these air equal. Well, let's look at the left hand side. So let's call U N the end partial some of the left hand side. So the left hand side. We know that we can always rewrite the whole sum as the limit of partial sums. So you and usually you see a written in this form, the whole son from one to infinity equals the limit of s end. So here, in our case, we have C and this is the entire sum going all the way up to infinity. Starting point might be one. Maybe not. But since it's not given much, just call it something. Let's call it one we can rewrite. This is a limit of the partial sums here, but still, actually, let me not call it s because we're using you. This's UK, The partial sums. So let's rewrite that partial some. So here we can rewrite UK as C A one si a Tu all the way up to see a k k partial some. This's just by definition of uk of here partials on then. Since I'm just dealing with a finite song, I could go ahead and factor out that see, And then I could also pull the si in front of the limit. This's just by one of your limit properties. So this is just using, ah, limit their, um, when you first learned about limits and then here let's go to the next page a one all the way up to a K. However, this now is just sn where's on the right hand side? We had this term here and then we're calling us in the sum of a one, as usual, up to an and then Therefore we could go ahead and replace The limit of SN was just the entire sum of a N, and that's exactly what we wanted to prove. So this is our final answer. The whole argument
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