divergence - Divergence of X n =2 1 n ln n As part of a...

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Unformatted text preview: Divergence of X n =2 1 n ln n As part of a homework question from 11.3 number 21, we know that n =2 1 n ln n diverges. Lets investigate how slow this diverges. (In fact, this series converges SLOWER than the Harmonic series.) Let S n = 1 2ln2 + 1 3ln3 + 1 4ln4 + + 1 n ln n and let f ( x ) = 1 x ln x . From the homework question, we know that f ( x ) is continuous, decreasing, positive and that a n = f ( n ). (By we know it is meant that YOU have done YOUR homework and verified this and thus we know.) Since the series diverges and goes off to infinity we can ask what k value will guarantee that S k is bigger than some other value. Lets say 10, i.e. we want to find the value of k for which S k > 10 is guaranteed. From the proof of the Integral Test, we saw that R n 2 f ( x ) dx S n- 1 . Thus, we want to find the value of k so that the following holds: 10 < Z k 2 f ( x ) dx S k- 1 So 10 < Z k 2 f ( x ) dx = ln(ln( k ))- ln(ln(2)) = ln ln k ln2 ! Hence k > 2 e 10 How big is that? Well, if we had a computer that could add two numbers in a nanosecond and we had the computer computing the sum and letting it run for the ENTIRE age of the known universe which is about 14 Billion years, the sum would STILL NOT REACH the number 10. Talk about slow divergence!!! The approximation of the number k and the age of the universe are on the next page. By the way, try typing 2 e 10 on your calculator. My calculator spit out an error. I had to use some high powered math software to even compute the value. We will now compare the age of the Universe in Nanoseconds to the value of k . Age of the Universe in Nanoseconds: 441 504 000 000 000 000 000 000 000 (about 14 Billion Years) Value of k = 4 235 480 664 700 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000...
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divergence - Divergence of X n =2 1 n ln n As part of a...

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