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Suppose 1/2 + 1/3 + 1/4 + ... + 1/x > 2.

 

What is the smallest possible value for x?

 Apr 2, 2021
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Just listing out the terms, we get:

 

1/2+1/3=5/6

1/2+1/3+1/4=13/12

1/2+1/3+1/4+1/5=77/60

1/2+1/3+1/4+1/5+1/6=87/60

1/2+1/3+1/4+1/5+1/6+1/7=223/140

1/2+1/3+1/4+1/5+1/6+1/7+1/8=481/280

1/2+1/3+1/4+1/5+1/6+1/7+1/8+1/9=4609/2520

1/2+1/3+1/4+1/5+1/6+1/7+1/8+1/9+1/10=4861/2520

1/2+1/3+1/4+1/5+1/6+1/7+1/8+1/9+1/10+1/11=55991/27720(first number greater than 2)

 

From this, we can see the smallest integer value of x is 11. However, this way of listing out terms is slow, and if anybody else has an algebraic way, it would be much appreciated.

 Apr 2, 2021

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