2.4 The Harmonic Series (continued)
We have not yet shown that our increasing sequence actually does approach a well-defined value. Using the Archmimedean understanding, we would have to show that there actually is a number with the property that for any real number x larger than , all values of stay below this x, and for any real number x below , then once n is large enough, all values of will be above x. How we possibly show this?
Let us consider another sequence defined by
We see that is just a little larger than , 1/n larger to be exact. While the are getting larger as n increases, the are getting smaller. This also can be seen geometrically. The difference between one value of this sequence and the next is
This is the area from x = n to x = n+1 that lies below the graph of y = 1/x and above the horizontal line y = n+1.
The value of is simply that value pinned in by these two converging sequences, one coming in from below and the other from above, and getting arbitarily close to each other.Anything larger than will eventually be larger than one of the . Anything smaller than will eventually be smaller than one of the .
There is only one small problem remaining. How do we know that there is a number thatis strictly less than every and strictly greater than every ?
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