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Explaining mathematical expressions with GPT-3

I wanted to see if GPT-3 could explain a mathematical expression for me, as a follow-on from Using GPT-3 to explain how code works. I grabbed an example from this page of the GitHub documentation.

Here's my conversation with GPT-3 as plain text - except I added > to the lines that I entered to differentiate them from the GPT-3 output:

> Explain this:
>
> $$\left( \sum_{k=1}^n a_k b_k \right)^2 \leq \left( \sum_{k=1}^n a_k^2 \right) \left( \sum_{k=1}^n b_k^2 \right)$$

This is the Cauchy-Schwarz inequality.

> Explain the formula syntax step by step, using $$ wrapped around the formula steps:

$$\left( \sum_{k=1}^n a_k b_k \right)^2 \leq \left( \sum_{k=1}^n a_k^2 \right) \left( \sum_{k=1}^n b_k^2 \right)$$

This is the Cauchy-Schwarz inequality.

$$\left( \sum_{k=1}^n a_k b_k \right)^2$$

This is the sum of all products of corresponding elements in the vectors a and b, squared.

$$\leq \left( \sum_{k=1}^n a_k^2 \right) \left( \sum_{k=1}^n b_k^2 \right)$$

This is saying that this sum is less than or equal to the product of the sums of the squares of each vector.

The reason I said "using $$ wrapped around the formula steps" was to get the output to be produced in the syntax that GitHub Markdown understands. It worked! Here's the rendered version of my conversation:


Explain this:

$$\left( \sum_{k=1}^n a_k b_k \right)^2 \leq \left( \sum_{k=1}^n a_k^2 \right) \left( \sum_{k=1}^n b_k^2 \right)$$

This is the Cauchy-Schwarz inequality.

Explain the formula syntax step by step, using $$ wrapped around the formula steps:

$$\left( \sum_{k=1}^n a_k b_k \right)^2 \leq \left( \sum_{k=1}^n a_k^2 \right) \left( \sum_{k=1}^n b_k^2 \right)$$

This is the Cauchy-Schwarz inequality.

$$\left( \sum_{k=1}^n a_k b_k \right)^2$$

This is the sum of all products of corresponding elements in the vectors a and b, squared.

$$\leq \left( \sum_{k=1}^n a_k^2 \right) \left( \sum_{k=1}^n b_k^2 \right)$$

This is saying that this sum is less than or equal to the product of the sums of the squares of each vector.

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