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Summing inverses (Posted on 2004-08-19) Difficulty: 3 of 5
What's the limit, as n→∞, of 1/(n+1)+1/(n+2)+1/(n+3)+...+1/(2n)?

See The Solution Submitted by Federico Kereki    
Rating: 4.0000 (5 votes)

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re(3): Reason why sum of limits is not same as limit as sums | Comment 14 of 18 |
(In reply to re(2): Reason why sum of limits is not same as limit as sums by np_rt)

"So in summary, the theorem that says that the sum of limits is the same as the limit of the sums does not hold for an infinite number of functions."

The sum, being one function of n, is just one function, so what is meant is and infinite number of terms.  However, it does hold for some functions that involve an infinite number of terms, such as

sin(x)= x - x^3/3! + x^5/5! - x^7/7! + ...

Each term smoothly approaches the value it has for a given x, and the total of the limits equals the limit of the total.

The way this differs from the function in the problem is best shown by individual values:

f(3) = 1/4 + 1/5 + 1/6

f(4) = 1/5 + 1/6 + 1/7 + 1/8

f(5) = 1/6 + 1/7 + 1/8 + 1/9 + 1/10

In this instance we can't keep track of "a given term" to find its limit.  The number of terms keeps increasing.

The same could conceivably happen in a series that starts out as infinite (the way the Maclaurin series for sine starts out as infinite, but differently defined), so long as there was not a 1-to-1 correspondence of terms from one iteration of the series to the next.  But the problem lies in the lack of correspondence of terms in successive valuations--not in the series being infinite.

  Posted by Charlie on 2004-08-20 13:48:41
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