Some guesses

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Even though we lack Gauss's genius, we can recognize certain patterns. One pattern is that if $\dot{x}$ is a function that gets bigger and bigger, it seems like will be a function that grows even faster than $\dot{x}$ . In the example of $\dot{x}=n$ and $\dot{x}=(n^2+n)/2$ , consider what happens for a large value of n, like 100. At this value of , $\dot{x}=100$ , which is pretty big, but even without pawing around for a calculator, we know that is going to turn out really really big. Since is large, n^2 is quite a bit bigger than , so roughly speaking, we can approximate $x\approx n^2/2=5,000$ . 100 may be a big number, but 5,000 is a lot bigger. Continuing in this way, for n=1000 we have $\dot{x}=1000$ , but $x\approx 500,000$ --- now has far outstripped $\dot{x}$ . This can be a fun game to play with a calculator: look at which functions grow the fastest. For instance, your calculator might have an x^2 button, an e^x button, and a button for (the factorial function, defined as $x!=1\cdot 2\cdot ...\cdot x,\textrm{e.g}.,4!=1\cdot 2\cdot 3\cdot 4=24$ ). You'll find that 50^2 is pretty big, but $e^{50}$ is incomparably greater, and 50! Is so big that it causes an error.

All the and $\dot{x}$ functions we've seen so far have been polynomials. If is a polynomial, then of course we can find a polynomial for $\dot{x}$ as well, because if is a polynomial, then x(n)-x(n-1) will be one too. It also looks like every polynomial we could choose for $\dot{x}$ might also correspond to an that's a polynomial. And not only that, but it looks as though there's a pattern in the power of . Suppose is a polynomial, and the highest power of it contains is a certain number - the “order” of the polynomial. Then $\dot{x}$ is a polynomial of that order minus one. Again, it's fairly easy to prove this going one way, passing from to $\dot{x}$ , but more difficult to prove the opposite relationship: that if $\dot{x}$ is a polynomial of a certain order, then must be a polynomial with an order that's greater by one.

We'd imagine, then, that the running sum of $\dot{x}=n^2$ would be a polynomial of order 3. If we calculate x(100)=1^2+2^2+...+100^2 on a computer spreadsheet, we get 338,350, which looks suspiciously close to 1,000,000/3. It looks like n^3/3+... , where the dots represent terms involving lower powers of such as n^2 . The fact that the coefficient of the n^3 term is 1/3 is proved in Problem 1.21.

Example

Figure 1.5 A pyramid with a volume of

1^2+2^2+3^2
Figure 1.5 shows a pyramid consisting of a single cubical block on top, supported by a $2\times 2$ layer, supported in turn by a $3\times 3$ layer. The total volume is , in units of the volume of a single block.

Generalizing to the sum x(n)=1^2+2^2+...+n^2 ,and applying the result of the preceding paragraph, we find that the volume of such a pyramid is approximately (1/3)Ah , where A=n^2 is the area of the base and h=n is the height.

When is very large, we can get as good an approximation as we like to a smooth-sided pyramid, and the error incurred in $x(n)\approx (1/3)n^3+...$ by omitting the lower-order terms ... can be made as small as desired.

We therefore conclude that the volume is exactly (1/3)Ah for a smooth sided pyramid with these proportions.

This is a special case of a theorem first proved by Euclid (propositions XII-6 and XII-7) two thousand years before calculus was invented.

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Some guesses

Example