The plot shows a comparison between the exact function $e^{\sqrt{x}}$ and the truncated Taylor series (up to seven terms) in the range $0 \leq x \leq 5$.

Truncation Error at $x = 5$:

• The exact value of $e^{\sqrt{5}}$ is approximately $9.356$.
• The value of the truncated Taylor series at $x = 5$ is approximately $12.010$.
• The truncation error at $x = 5$ is approximately $2.653$.

This error indicates the difference between the exact value and the approximation using the first seven terms of the Taylor series.

Would you like further details or have any questions?

Here are 5 related questions:

1. How is the convergence of the Taylor series for $e^{\sqrt{x}}$ affected as $x$ increases?
2. How do higher-order terms affect the truncation error?
3. Can you generalize the pattern for the Taylor expansion of other functions involving square roots?
4. How does the radius of convergence of a Taylor series impact its accuracy?
5. How would the Taylor expansion change if centered at a different point?

Tip: The convergence of a Taylor series is faster when approximating near the point where the series is centered (in this case, around $x = 0$).