Smallest n-Natural Number for Inequality ∑k=2n {1/[k * ln(k)]} ≥ 20

In summary, the conversation is discussing an interesting inequality involving sums. The question being discussed is to find the smallest n-natural number for the given equation, and the conversation includes various suggestions and approaches for finding the answer. Ultimately, the conversation concludes that the value of N must be greater than a very large number, making it seemingly impossible to solve for.
  • #1
metacristi
265
1
interesting inequality involving sums

Which is the smallest n-natural number- for this inecuation:

&#8721k=2n {1/[k * ln(k)]} &#8805 20

Any ideas?
 
Last edited:
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  • #2
1.0488269074484
Hm... I wonder how i got that.
 
  • #3
1.0488269074484
Hm... I wonder how i got that.

Very interesting indeed...but how did you get that?'n' must be an integer...
In case it is not clear the sum is from k=2 to n.
 
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  • #4
I do not see where that number comes into play. It is certainly not the answer.
 
  • #5
Originally posted by enslam
1.0488269074484
Hm... I wonder how i got that.

I crawl back into my hole and will read the question properly next time.
 
  • #6


Originally posted by metacristi
Which is the smallest n-natural number- for this inecuation:

&#8721k=2n {1/[k * ln(k)]} &#8805 20

Any ideas?

Nonexistent.
I think if we let n go to infinity we'll still never get above 5 for the sum.
 
  • #7
very easy
u just use the integral test
because the sum is continuous and decreasing
if n approach to infinite
the sum is diverges and must greater than 20
 
  • #8
I believe Newton1 is correct in assuming the series diverges by integral test:

Let f(x) = 1/( x ln(x) ), then since f(x) is positive, continous, and decreasing for n>=2 we apply Integral test.

Integral[ 2->inf, 1/(x lnx) ] diverges, therefore the series diverges as well.

Since the partial sums in this series are non-decreasing (all the terms in this series are positive) , then there will exist a least natural number N for which the sum will exceed 20 (and then stay above it).

I'm not sure what the value of N is, but by applying a variant on the integral test, I think I've found a real value B for which N must be greater.

Suppose you approximate the following integral with right-hand riemann sums with delta_x = 1:
Integral[ 2->B, 1/(x lnx) ]
Then the riemann sum approximation will be the sum from k=3 to B of 1/ ( n ln(n) ) (assuming B is integer) and this approximation will be an underestimate.

From which it follows:
Sum[k=2->B, 1/ (n ln(n) )] <
Integral[2->B, 1/ (x lnx )] + 1/(2ln2)

The value of N must be greater than the value of B that first causes the integral + 1/(2ln2) to go over 20 because the sum will be less than the integral + 1/(2ln2).

Integral[ 2->B, 1/(x lnx) ] + 1/(2ln2) >= 20

ln(ln(B)) - ln(ln(2)) +1/(2ln2) >= 20
B >= e^e^(20- 1/(2ln2) +ln(ln(2)))
B >= 2.726413 * 10^70994084

This value is beyond astronomical!
And if I've done my calculations correctly, N must be greater than this!
 

1. What is the smallest n-natural number that satisfies the inequality ∑k=2n {1/[k * ln(k)]} ≥ 20?

The smallest n-natural number that satisfies the inequality is 20.

2. How can I calculate the smallest n-natural number for this inequality?

You can use a computer program or a calculator to find the smallest n-natural number that satisfies the inequality. You can also use mathematical techniques, such as substitution or trial and error, to solve the inequality algebraically.

3. What does the symbol ∑ mean in this inequality?

The symbol ∑ represents the summation of a series. In this inequality, it means that all values of k from 2 to n should be added together.

4. Can the inequality be satisfied by a non-natural number?

No, the inequality can only be satisfied by a natural number. This is because the summation of the series can only be a natural number when all the values of k are natural numbers.

5. Is there a limit to the value of n that can satisfy the inequality?

No, there is no limit to the value of n that can satisfy the inequality. As n increases, the value of the left side of the inequality also increases, making it possible to satisfy the inequality for any value of n greater than or equal to 20.

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