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Dealing with a recent question I spotted a very nice exercise for Calc-2 students, i.e. to find the mistake in the following lines.

Lemma 1. For any $n\in\mathbb{N}$, we have: $$ \int_{0}^{1} x^n\left(1+(n+1)\log x\right)\,dx = 0. $$ Lemma 2. For any $x\in(0,1)$ we have: $$ \frac{1}{1-x}=\sum_{n\geq 0}x^n,\qquad \frac{\log x}{(1-x)^2}=\sum_{n\geq 0}(n+1) x^n\log(x). $$ By Lemmas 1 and 2 it follows that: $$\begin{eqnarray*}(\text{Lemma 1})\quad\;\;\color{red}{0}&=&\int_0^1 \sum_{n\geq0} x^n\left(1+(n+1)\log x\right)\,dx\\[0.2cm](\text{Lemma 2})\qquad&=&\int_0^1 \left(\frac{1}{1-x} + \frac{\log x}{(1-x)^2}\right)\,dx\\[0.2cm](x\mapsto 1-x)\qquad&=&\int_0^1 \left(\frac{1}{x}+\frac{\log(1-x)}{x^2}\right)\,dx\\[0.2cm](\text{Taylor series of }x+\log(1-x))\qquad&=&-\int_0^1 \frac{1}{x^2} \sum_{k\geq2}\frac{x^k}k \,dx\\[0.2cm](\text{termwise integration})\qquad&=&-\sum_{k\geq 2} \frac{1}{k(k-1)}\\[0.2cm](\text{telescopic series})\qquad&=&-\sum_{m\geq 1} \left(\frac{1}{m}-\frac{1}{m+1}\right)=\color{red}{-1}. \end{eqnarray*}$$

Now the actual questions: were you able to locate the fatal flaw at first sight?
Do you think it is a well-suited exercise for Calculus-2 (or Calculus-X) students?

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    $\begingroup$ I am not familiar with both the content and the prerequisites of a "Calculus-2" class. Could you give a brief outline of what the students should learn and what they should know already? $\endgroup$ – Christian Jul 5 '16 at 9:04
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    $\begingroup$ Not suitable if students do not know "What conditions make $\int f_n\rightarrow \int f$?" As far as I know that question is not answered in most (lower division) calculus class I took/taught/knew. $\endgroup$ – user2139 Jul 5 '16 at 9:09
  • $\begingroup$ Seems like a good question for a real analysis course rather than a standard calculus course. $\endgroup$ – John Coleman Jul 5 '16 at 15:45
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[...]were you able to locate the fatal flaw at first sight?

Given the context I was a priori quite certain it would be an issue in interchanging limits. What else could it be? You would not make an index error or something like this. Yet, this is a bit 'cheating' though. It is a nice problem.

Do you think it is a well-suited exercise for Calculus-2 (or Calculus-X) students?

I would likely not give this as a regular exercise, and certainly not as an exam question, as the 'variance' is too large (it can be quick and easy, or one can spend enormous amounts of time based in part on getting lucky to start in the right way).

However, as some bonus or optional exercise. Sure, as long as the students have a grasp of the individual bits and pieces, that is Taylor series etc. Maybe I would even consider it before discussing interchanging integrals and limits. Then it can serve as an example that motivates the need. In this form one might even present it in a lecture, not as exercise.

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