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Find all positive values of $ b $ for which the series $ \sum_{n = 1}^{\infty} b^{\ln n} $ converges.

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since $b$ has to be positive also, so $\sum_{n=1}^{\infty} b^{\ln n}$ will converge as long as $0<$$b<\frac{1}{e}$

Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 3

The Integral Test and Estimates of Sums

Sequences

Series

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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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in this problem, we'd like to find all the values of B there allow the Siri's to converge. So what we'LL do is just start off by playing with this and term here. So recall that weaken right b is e to the Ellen B. So rewrite bee in this form and then properties of exponents says when you raise a power to another power you got it won't supply those powers. So here, let me go ahead. And actually, and because of that fact, let me just go ahead and change the role of and then B And the reason again is because of this fact Here, thes two quantities are equal because they're both equal to this term here. So here I am, just switching the exponents using this fact. And then now, using this again, I can rewrite the term in the print. Decease is just end. And then I get n Ellen be and we let's write This is one over and negative. Ellen, be so we've shown here Is that the original Siri's Khun B. Ren as a P series and this will make it easier for us to answer the question. So this is Ahh P Siri's with P negative ln b and remember, p series converges if and only if if peace is bigger than one. So we need negative Ellen be to be larger than one and then we'LL just solve this for being Some will supply both sides by negative one and take e on both sides. The reason this is true is because if X is less than why than e X is less than either the y so that justifies this fact here. And then we can rewrite the left hand side again by using This is B and so we have B is less than one over he So here we wanted also we wanted positive values. If you go back to the original wording in the problem, so here we should also throw in the condition that these larger than zero hey, and that's your final answer

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