I introduce a utility function `Y_(f,x,`

`)`

, the
purpose of which is to compute the value of the function at .
In many utility files that come along with the software the trick to achieve
this is to compute the
limit of the expression at . However, I prefer to
use the `ITERATE`

command in the following way:

#1: Y_(f,x,alpha) := ITERATE(f,x,alpha,1)The utility function can be used to compute the convolution product of two functions

#2: CONV(f,g,x):=INT(Y_(f,x,t)*Y_(g,x,x-t),t,-inf,inf)The probability density function of adding

#3: DEN_SUM(n,f,x):=IF(n=1,f,CONV(DEN_SUM(n-1,f,x),f,x))The standard uniform distribution is available in Derive.

#4: U(x):=CHI(0,x,1)

Below you see a screen dump in which I compute `DEN_SUM(n,U(x),x)`

for `n=1,2,3`

.

In Figure 7.2 you see the results for adding **4** and
**5** times. The best result is obtained after applying the `Factor`

command once or twice on the intermediate reult of simplifying
`DEN_SUM(4,U(x),x)`

and `DEN_SUM(5,U(x),x)`

.

You may recognize Pascal's triangle and conjecture the same general formula as I do:

In Derive I author this formula as the function `UN(n,x)`

:

#18: UN(n,x):=1/(2(n-1)!)*SUM((-1)^k*COM(n,k)*SIGN(x-k)*(x-k)^(n-1),k,0,n)

Now the proof by induction can start. First the check that
`UN(1,x)=U(x)`

.

#19: U(x) - UN(1,x) = 0Supposing that the formula holds for

`n`

I can compute
#20: CONV( UN(n,x), U(x) )The result of simplification is partly shown on line #21 in Figure 7.3.

**Figure 7.3:** Proof by induction

What are in short the steps I take above? Well,
Fiddling with the `F3`

key and with manual changes in the `Author`

and
`Build`

command lines the intermediate result is rewritten as the formula in line
#22. The steps are omitted but I combine integrals and sums, and I interchange summation and integration.
In Figure 7.4 below you see that the integration interval is restricted to on the basis of
elementary properties of the `SIGN`

function.
Derive can compute the integral and with a bit of help in the handling of sums --- intermediate results are
in line #25 and #26 --- I find the formula of line #27:

I can verify with Derive that this is indeed equal to `UN(n+1,x)`

:

**Figure 7.4:** Proof by induction (continued)

**Figure 7.5:** Proof by induction (continued)

Sun Apr 23 10:32:10 MDT 1995