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JH
Numerade Educator

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Problem 76 Hard Difficulty

Find the value of c such that
$ \displaystyle \sum_{n = 0}^{\infty} e^{nc} = 10 $

Answer

$c=\ln \frac{9}{10}$

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AZ

Aurang Z.

April 26, 2021

Let F:R^4?R^3 be a linear transformation with range {(t,0,0) | t? R} What is the nullity of F?

Video Transcript

let's find the value of C search. That there's some over here could not only converges but equals ten. So this some over here on the left hand side. We can actually rewrite this so that its geometric it is geometric, but it might not be obvious the way that it's written. So here are pull out thiss and here, and the reason you can do this is just your laws of exponents. So this is geometric with our equals either the sea that that's the thing that's being raised to the end power and this thing won't converse for all C but will converse for some sea because here absolute value are it's either the scene and you could just write that. Is this because either the exes always positive, so you don't have to worry about a negative sign? You said that to be less than one, because that's when geometric series convergence converges. If absolute value bars less than one and here you can solve this for seed by just taking the natural log on both sides, so seize less than zero. Then the Siri's will convert. But that's not enough because we want to know also when the sum equals ten. So assuming that sees less than zero, we know that the sum of the two geometric series. So this assumes that emerges, so sees less than zero. Then this will equal the first term of the series that you get by plugging in and equal zero over one minus R seasonings on this case, the first term plugging an equal zero and you get even the zero, which is one come on, one minus r, which is either the sea and we want this to be equals a ten on one hand. This is the sum of the Siri's because she a metric that's where the fractions coming from on the left side. On the other hand, we wantto find see such that this equals ten. So that's why we set the tent on the really inside. So all we're going to do now is go to the next page here and just solve this new equation, Percy. And that will be our final answer. One over one minus. Either the sea equals ten. This is equivalent. You could just float both sides here one minus e to the C equals one over his head. simplify this that's nine over ten equals either the sea and then I would just take lot on both sides. And that's our final answer. And also here we can see that this number C is a negative number, and that was a requirement for C C ten organ to converge. This see how to be less than zero. And so there actually is a sea value less than zero, and it's natural log of nine over ten, and that's our final answer.