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Find the limit of the sequence$ \left\{ \sqrt 2, \sqrt{2\sqrt2}, \sqrt{2\sqrt{2\sqrt2}}, \cdot \cdot \cdot \right\} $

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Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 1

Sequences

Series

Harvey Mudd College

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.

01:01

Find the limit (if possibl…

01:42

Determine whether the sequ…

00:59

Calculate the limit.$\…

00:32

Find the indicated limit.<…

Let's find the limit of this sequence. If we look at the first term, weaken right, This is too too. The one half the second star weaken right. This is two times two to the one half to the one half power. So simplifying this we have to do the three forints. And for the third term, let's do this as well. This is two times two to the three fourths to the one half. So that's two to the seven over eight. And it looks like we may have enough here to see the pattern. So the end of term is given by a N equals two. And now what's the exponents? So we had one half three fourth, seven eight. So notice that the numerator is always one less than the denominator and noticed that the denominator is of the form to the end. So this means we have to to the end and then numerator is one less. So you go back up there, subtract one, and so we have limit, take a luminous and goes to infinity. So now we have to to the end power minus one groups a minus one should not be in the exponents. It should be to the whole thing and then to the end. Now we'LL use the fact that this power function here, this exponential function excuse me is continuous so I can go in and take this limit and right in the exponents. Yeah, and then either by doing some algebra or using low Patel's rule, you can see that this exponents goes toe one. So this is just to the one which equals two. And this is the limit of the sequence to, and that's our final answer.

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