# Using your Head is Permitted

## March 2010 solution

If one allows *u* to be zero then a
simple example of one such *f* is *f*(*x*)=1/*x*.
The series *T*_{f|a}(*x*), in this case,
is the sum from *i*=0 to infinity of
*a*^{-i-1}(*x*-*a*)^{i} =
(*x*/*a*-1)^{i}/*a*.

This is a geometric series that converges when
|*x*/*a*-1|<1, meaning when *x* ∈ (0,2*a*).

Therefore, for any *d* we can pick *a*=*d*/2,
*u*=0.

This solution, however, uses *u*=0, and the riddle explicitly asks for
a positive *u*. An example with a positive *u* would be

*f*(*x*)=e^{-1/(x-1)} for *x*>1 and
*f*(*x*)=0 for *x*≤1.

In this function, *T*_{f|a}(*x*) equals
*f*(*x*) in the range (1,2*a*-1) for *a*>1. Therefore,
we can choose *a*=*d*/2+1 for any *d*.

(My thank-yous to Christian Blatter for pointing out to me that *u*=0
violates the wording of the riddle.)

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