Cauchy condensation examination - Exponential function description

I simply ended up reviewing the Wikipedia write-up on the Cauchy condensation test. I recognize the trapezoidal sight, yet evidently "the 'condensation' of terms is similar to a replacement of a rapid function". Can any person clarify what is suggested by this?

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2019-05-04 17:40:14
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Answers: 2

Explanation is fairly uncomplicated. Take into consideration indispensable $\int f(x)dx$. After adjustment of variable $x=2^t$ it comes to be $const\cdot\int 2^t f(2^t)dt$-- compare to $\sum 2^n f(2^n)$ from Cauchy condensation examination.

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2019-05-08 02:54:30
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" Condensation" due to the fact that the collection $a_n$ is pressed to a subseries $a_{2^n}$ lugging just the same details concerning convergence.

The evidence of the condensation examination is very easy, as Grigory M's solution shows and also whenever it functions so does a contrast of amount with indispensable, which is an extra basic suggestion. So you could normally ask yourself why this apparently unimportant convergence examination deserves its very own name, apart from the popularity of its intended developer. The factor it is preserved in books is that it is usually a hassle-free standard to make use of for collection when the terms or the partial amounts have some dependancy on $\log (n)$. The evidence of aberration of the harmonic collection is an instance.

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2019-05-08 02:32:38
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