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Is the 50th partial sum $ s_{50} $ of the alternating series $ \displaystyle \sum_{n = 1}^{\infty} (-1)^{n - 1} /n $ an overestimate or an underestimate of the total sum? Explain.

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underestimate since $\sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n}=s_{50}+\left(\frac{1}{51}-\frac{1}{52}\right)+\left(\frac{1}{53}-\frac{1}{54}\right)+\cdots \cdot,$ and the terms in parentheses are all positive.

Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 5

Alternating Series

Sequences

Series

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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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Is the 50th partial sum $s…

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Is the 50 th partial sum $…

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Alternating Series Determi…

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Find the partial sum.$…

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Find the partial sum.…

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A series $\sum_{n=1}^{\inf…

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Find the sum of the given …

is the fiftieth. Partial some and overestimate or underestimate for the total sum. So for all trading theories, we always have one of the following situations. So the even sub scripts are always less than the sum, which is always less than the odd sub scripts or the opposite is true. This is shrewd and saw the picture of each case. Here's your some and then you have, like, a swan three as five and so on. That's not going in the right direction. So let me go back and change that. Here's s one as three and so on. And then here you have us, too. As for oranges, have things what switched around a little bit as one history. So they're getting closer to us. But all the evens are on the right and all the odds are on the left. So here to find out what case we're in. So case Air Case B. So it's one or the other, not both. So let's find out which case we're in. So let's find that one as one is just one, and this will be an overestimate since, for example, we can just the way to answer this the way to find out whether it's an overestimate, as one I'm saying is just by comparing it test too, so as to is s one plus and then we have negative one to the two minus one over two. So that's one minus a half much is a half, so we can see that as one is bigger than as too. So we must be in case, eh? It's one or the other. So therefore, as one is overestimate. So this tells us that all the odds are in our overestimate. So the even numbers are the lower estimates. So since ESA fifty will fall on this side, fifty is even so as the Sun fifty is less than the entire sum. So that's not a five that's s there. So we don't know that with with a dollar sign to distinguish my my s is from fives. So that's a sum from one to infinity minus one to the end, minus one over end. So we see that this is an underestimate, since it's less than and that's our final answer

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