This is not homework. Problem 3-38 reads:
Let $A_{n}$ be a closed set contained in $(n,n+1)$. Suppose that $f:\mathbb{R}\rightarrow \mathbb{R}$ satisfies $\int_{A_{n}}f=(-1)^{n}/n$ and $f(x)=0$ for $x\notin$ any $A_{n}$. Find two partitions of unity $\Phi$ and $\Psi$ such that $\sum_{\phi\in\Phi}\int_{\mathbb{R}}\phi\cdot f$ and $\sum_{\psi\in\Psi}\int_{\mathbb{R}}\psi\cdot f$ converge absolutely to different values.
A few observations: First, $n\ge 1$. Second, Spivak uses what he calls an extended integral, whose definition and relations with the usual integral can be found on p.65, which can be found here or here.
It may be helpful to have an example of such a function in mind. Let $A_{n}=$ closed interval of length $1/2n$ centered at the point $(2n+1)/2$. Clearly $A_{n}\subset(n,n+1)$. then define $f(x)=\begin{cases} \hphantom{-}2& \text{if $x\in A_{n}$ for $n$ even}\\ -2& \text{if $x\in A_{n}$ for $n$ odd}\\ \hphantom{-}0& \text{otherwise}. \end{cases}$
A possible approach: Let $a_{n}=(-1)^{n}/n$. Since $\sum_{n}a_{n}=\alpha\in\mathbb{R}$ but the convergence is conditional, then for any $\beta\not=\alpha$ there is a rearrangement $\{b_{n}\}$ of the sequence $\{a_{n}\}$ such that $\sum_{n}b_{n}=\beta$.
Now, we form a family of open sets $\{U_{n}\}$, where $U_{n}$ is the union of $n$ intervals $(k,k+1)$, each corresponding to a term of the $n$-th partial sum of $\sum_{n}a_{n}$. We form a similar family $\{V_{n}\}$ looking at the partial sums of $\sum_{n}b_{n}$. E.g., if we let $\{b_{n}\}=\{-1,1/2,1/4,-1/3,1/6,1/8,-1/5,\ldots\}$ we have $V_{3}=(1,2)\cup(2,3)\cup(4,5)$ while since $\{a_{n}\}=\{-1,1/2,-1/3,1/4,\ldots\}$ we have $U_{3}=(1,2)\cup(2,3)\cup(3,4)$.
If we slightly fatten-up the $U_{n}$ (resp. the $V_{n}$) we form open covers $\mathcal{U}$ (resp. $\mathcal{V}$) of all the reals greater or equal than 1 without ading points where $f$ in non-zero. My heart tells me that partitions of unity $\Phi$ and $\Psi$ subordinate to $\mathcal{U}$ and $\mathcal{V}$, respectively, will be the desired one. But alas I am lost!
Does any one know how to show that the aforementioned partitions of unity are the desired ones?
Other possible approaches to the solution are also welcomed.
In addition to two posts linked above, related issues with other problems and statements about integration in Spivak´s book can be found here and here.