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Help me to solve 3 and 4 with the graphics

The graphs above illustrate the comparison between the actual functions and their corresponding Taylor polynomials for \(n = 5\):

1. **For \(a^x\) with \(a = 2\)** (left plot), the Taylor series approximation closely follows the exponential curve, particularly near \(x = 0\), but diverges for larger \(x\).

2. **For \(\frac{1}{1+x}\)** (right plot), the Taylor series accurately represents the function near \(x = 0\), but starts to deviate significantly as \(x\) approaches \(-1\) (where the function has a vertical asymptote) and for large \(x\).

These visualizations help illustrate how Taylor series approximate functions and where they start to lose accuracy as \(x\) moves away from the expansion point.

Would you like a more detailed explanation or further questions about these concepts?

Here are five related questions to deepen understanding:
1. How does the convergence of a Taylor series depend on the degree \(n\)?
2. Why does the Taylor series for \(\frac{1}{1+x}\) diverge near \(x = -1\)?
3. How would the Taylor series for \(a^x\) behave for different values of \(a\)?
4. What happens to the accuracy of the Taylor polynomial if we increase \(n\)?
5. Can you explain why the Taylor series for \(1/(1+x)\) is geometric?

**Tip:** Taylor series are best near the point of expansion (usually \(x = 0\)). Farther away, higher degrees are needed for accuracy.

Help me to solve 3 and 4 with the graphics.

Solve exercises 3 and 4 from the Taylor series expansion section, focusing on the functions a^x and 1/(1+x). Learn how to derive the series, understand polynomial approximations, and see graphical comparisons. Suitable for undergraduate students studying calculus.

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