Solving Summation Problem: Show f(n) is Not an Integer

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In summary, the conversation discusses the function f(n) = 1/2 + 1/3 + ... + 1/n and attempts to show that it is not an integer for any positive integer n. Different methods are suggested, including rearranging the terms and using induction, but no concrete solution is found.
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VeeEight
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Homework Statement



Let f(n) = 1/2 + 1/3 + ... + 1/n
Show that f(n) is not an integer for any positive integer n


The Attempt at a Solution



I think that rearraning/breaking down the statement might be easier than applying a theorem since it seems like a simpler problem. Simply arranging the terms got pretty messy so I think the best method is to try to find an integer C such that C x f(n) is not a integer for any value of n. I tried different ways of computing such an integer C, such as taking C to be (n-1)! or something similar for that the summation will give a bunch of integers plus one term that is not an integer but I failed to find such a value. Hopefully someone can help me out over here. Thanks.
 
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  • #2
I'm a little rusty on this stuff but couldn't we solve this using induction?
 
  • #3
The funny thing is [tex]f(1)=1/1[/tex] and [tex]1[/tex] is an integer. But, aside from that, IDK.
 

1. What is a summation problem?

A summation problem is a mathematical problem that involves adding up a series of numbers. It is often represented using the summation symbol (∑) and an expression that tells you how to calculate each term in the series.

2. How do you show that f(n) is not an integer in a summation problem?

To show that f(n) is not an integer in a summation problem, you need to find a value of n that makes f(n) a non-integer. This can be done by plugging in different values of n and checking if the result is a whole number or not.

3. What is the significance of proving that f(n) is not an integer in a summation problem?

Proving that f(n) is not an integer in a summation problem is important because it helps to determine the convergence or divergence of the series. If f(n) is not an integer for all values of n, then the series is likely to be divergent.

4. Can you use induction to prove that f(n) is not an integer in a summation problem?

Yes, you can use mathematical induction to prove that f(n) is not an integer in a summation problem. This involves showing that the statement is true for a base case (usually n=1) and then proving that if it is true for n=k, it is also true for n=k+1.

5. Are there any specific techniques or strategies for solving summation problems?

Yes, there are several techniques and strategies that can be helpful in solving summation problems. These include breaking the summation into smaller parts, using known summation formulas, and manipulating the series to make it easier to work with. It is also important to carefully consider the properties of the series, such as convergence or divergence, before attempting to solve it.

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