News flash: ChatGPT sometimes asserts false statements are true

Calculus, Technology 1 Minute

I feel like I’ve done my good deed for the day by uncovering an instance when ChatGPT claimed a “fact” from the secondary mathematics curriculum that is simply incorrect.

While using ChatGPT to do some brainstorming on a research project, I received the following response (as part of a bigger response):

Sadly, whenever ChatGPT asserts a fact without documentation, it behooves the user to double-check the “fact.” In this case, there is a pretty blatant sign error.

As taught in high school AP calculus, the Taylor series expansion for \ln(1+y) is

\ln(1+y) = \displaystyle \sum_{q=1}^\infty \frac{(-1)^{q-1}}{q} y^q,

and so

-ln(1-y) = \displaystyle -\sum_{q=1}^\infty \frac{(-1)^{q-1}}{q} (-1)^q y^q

= \displaystyle\sum_{q=1}^\infty \frac{(-1)^{2q}}{q} y^q

= \displaystyle\sum_{q=1}^\infty \frac{y^q}{q}

Said another way,

\displaystyle \sum_{q=1}^{A-1} \frac{y^q}{q} + \sum_{q=A}^\infty \frac{y^q}{q} = -\ln(1-y)

\displaystyle \sum_{q=A}^\infty \frac{y^q}{q} = -\ln(1-y) - \sum_{q=1}^{A-1} \frac{y^q}{q}

\displaystyle \sum_{r=0}^\infty \frac{y^{r+A}}{r+A} = -\ln(1-y) - \sum_{q=1}^{A-1} \frac{y^q}{q}

\displaystyle \sum_{r=0}^\infty \frac{y^{r}}{r+A} = \frac{1}{y^A} \left[ -\ln(1-y) - \sum_{q=1}^{A-1} \frac{y^q}{q} \right]

So that’s the correct answer. Notice that there is a minus sign in front of the sum on the right-hand side, not a plus sign.

So I asked ChatGPT to double-check. Here’s the first part of the response.

Lesson: You get what you pay for.

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Published by John Quintanilla

I'm a Professor of Mathematics and a University Distinguished Teaching Professor at the University of North Texas. For eight years, I was co-director of Teach North Texas, UNT's program for preparing secondary teachers of mathematics and science.

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