To expand $e^x \cdot \log(1 + y)$ up to the 3rd degree in terms of $x$ and $y$, we'll use the series expansions of $e^x$ and $\log(1 + y)$.

Step 1: Taylor series expansion of $e^x$

The Taylor series of $e^x$ around $x = 0$ is: $e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \mathcal{O}(x^4)$ So, up to the 3rd degree: $e^x = 1 + x + \frac{x^2}{2} + \frac{x^3}{6}$

Step 2: Taylor series expansion of $\log(1 + y)$

The Taylor series of $\log(1 + y)$ around $y = 0$ is: $\log(1 + y) = y - \frac{y^2}{2} + \frac{y^3}{3} + \mathcal{O}(y^4)$ So, up to the 3rd degree: $\log(1 + y) = y - \frac{y^2}{2} + \frac{y^3}{3}$

Step 3: Multiply the two expansions

Now, we multiply $e^x$ and $\log(1 + y)$ up to the 3rd degree: $e^x \cdot \log(1 + y) = \left(1 + x + \frac{x^2}{2} + \frac{x^3}{6}\right) \cdot \left( y - \frac{y^2}{2} + \frac{y^3}{3} \right)$ We'll expand this multiplication, keeping terms up to the 3rd degree.

1. Multiply $1$ by $\log(1 + y)$:

$1 \cdot \left( y - \frac{y^2}{2} + \frac{y^3}{3} \right) = y - \frac{y^2}{2} + \frac{y^3}{3}$

2. Multiply $x$ by $\log(1 + y)$:

$x \cdot \left( y - \frac{y^2}{2} + \frac{y^3}{3} \right) = xy - \frac{xy^2}{2} + \frac{xy^3}{3}$

3. Multiply $\frac{x^2}{2}$ by $\log(1 + y)$:

$\frac{x^2}{2} \cdot \left( y - \frac{y^2}{2} + \frac{y^3}{3} \right) = \frac{x^2 y}{2} - \frac{x^2 y^2}{4} + \frac{x^2 y^3}{6}$

4. Multiply $\frac{x^3}{6}$ by $\log(1 + y)$:

$\frac{x^3}{6} \cdot \left( y - \frac{y^2}{2} + \frac{y^3}{3} \right) = \frac{x^3 y}{6} - \frac{x^3 y^2}{12} + \frac{x^3 y^3}{18}$

Step 4: Collect all terms up to the 3rd degree

Now, combining all terms up to the 3rd degree: $e^x \cdot \log(1 + y) = y - \frac{y^2}{2} + \frac{y^3}{3} + xy - \frac{xy^2}{2} + \frac{xy^3}{3} + \frac{x^2 y}{2} - \frac{x^2 y^2}{4} + \frac{x^2 y^3}{6} + \frac{x^3 y}{6} - \frac{x^3 y^2}{12} + \frac{x^3 y^3}{18}$

This is the expansion of $e^x \cdot \log(1 + y)$ up to the 3rd degree.

Would you like further details or have any questions?

Here are 5 related questions for further exploration:

1. How does the radius of convergence affect the series expansion?
2. What happens when expanding to higher degrees (e.g., 4th or 5th degree)?
3. Can you explain the error term in Taylor expansions?
4. How do you derive the expansion for more complex functions?
5. How does a multivariable Taylor series expansion work?

Tip: To improve accuracy, always keep track of the highest degree terms needed when multiplying series expansions.