WEBVTT
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were given a series and were asked to determine whether
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this series is convergent or divergent. So the series
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is 1/5 plus 1/7 plus 1/9 plus 1 11 plus
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1 13 and so on. When I say and
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so on, what does this mean? That is
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, how would you continue find the next term in
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this some? Well, we see that we'd simply
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find the next odd number after 13, just 15
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and take the reciprocal. So, nurse, we
10
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can write this as the sum from and equals,
11
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Let's say, zero to infinity of one over five
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plus to end course. This could also be written
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as these some from n equals one to infinity of
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one over two n Plus three doesn't really affect our
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method either way. Now, let's take ffx be
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the function 1/2 X plus three. Now we know
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that F of X is non negative on the interval
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from one to infinity. Indeed, it's non negative
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for all positive values of X. We also know
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that F is mon atomically decreasing on the interval from
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one to infinity. This is because it's an inverse
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function. In fact, it's decreasing for all X
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such that the denominator is non zero and therefore it
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follows that from the integral test follows that our series
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converges if and only if the integral from one to
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infinity of one over two. X plus three.
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The X converges. The question is, does this
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interval converge well and to go from one to infinity
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of 1/2 x plus three d x. This is
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the same as the limit as T approaches infinity of
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the integral from one duty of 1/2 x plus three
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d x, and this is equal to taking anti
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derivatives limit as T approaches infinity of one half times
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the natural log of two X plus three from X
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equals one t and plugging in. This is the
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limit as T approaches infinity of one half times the
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natural laws of two t plus three minus one half
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times the natural log of Just see, that's two
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times one is two plus three is five. Of
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course you know that limit This T purchase infinity of
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the natural log of two T plus three is again
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infinity. This is infinity minus one half natural log
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of five. Which of course is just infinity and
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therefore the integral diverges. And so it follows that
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these series also diverges by the integral test.