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Numerade Educator

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Problem 12 Easy Difficulty

Determine whether the series is convergent or divergent.
$ \frac {1}{5} + \frac {1}{7} + \frac {1}{9} + \frac {1}{11} + \frac {1}{13} + \cdot \cdot \cdot $

Answer

Diverges
Hint: Use Integral test

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Video Transcript

were given a series and were asked to determine whether this series is convergent or divergent. So the series is 1/5 plus 1/7 plus 1/9 plus 1 11 plus 1 13 and so on. When I say and so on, what does this mean? That is, how would you continue find the next term in this some? Well, we see that we'd simply find the next odd number after 13, just 15 and take the reciprocal. So, nurse, we can write this as the sum from and equals, Let's say, zero to infinity of one over five plus to end course. This could also be written as these some from n equals one to infinity of one over two n Plus three doesn't really affect our method either way. Now, let's take ffx be the function 1/2 X plus three. Now we know that F of X is non negative on the interval from one to infinity. Indeed, it's non negative for all positive values of X. We also know that F is mon atomically decreasing on the interval from one to infinity. This is because it's an inverse function. In fact, it's decreasing for all X such that the denominator is non zero and therefore it follows that from the integral test follows that our series converges if and only if the integral from one to infinity of one over two. X plus three. The X converges. The question is, does this interval converge well and to go from one to infinity of 1/2 x plus three d x. This is the same as the limit as T approaches infinity of the integral from one duty of 1/2 x plus three d x, and this is equal to taking anti derivatives limit as T approaches infinity of one half times the natural log of two X plus three from X equals one t and plugging in. This is the limit as T approaches infinity of one half times the natural laws of two t plus three minus one half times the natural log of Just see, that's two times one is two plus three is five. Of course you know that limit This T purchase infinity of the natural log of two T plus three is again infinity. This is infinity minus one half natural log of five. Which of course is just infinity and therefore the integral diverges. And so it follows that these series also diverges by the integral test.