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The problem is asking for the error in calculating \( x_{10} \) when using an approximate value of \( x_0 = 1.001 \) instead of \( x_0 = 1 \), based on a recursive sequence.

Given:
\[
x_0 = 1, \quad x_n = 10x_{n-1} - \frac{9n+10}{n^2+n}, \quad n=1,2,\dots
\]
And you're asked to evaluate the stability of the calculation when using \( x_0 \approx 1.001 \).

### Steps:
1. **Define the exact recursive sequence:** You start with \( x_0 = 1 \), then apply the recursive formula for \( n=1, 2, ..., 10 \).
   
2. **Approximate value calculation:** Recalculate the sequence with \( x_0 = 1.001 \) and propagate it to \( x_{10} \).

3. **Error evaluation:** Compute the difference between the two values of \( x_{10} \) obtained from \( x_0 = 1 \) and \( x_0 = 1.001 \).

Let me calculate the values for you and check the error at \( x_{10} \).The calculation yields the following results for \( x_{10} \):

- Using the exact value \( x_0 = 1 \), \( x_{10} \approx 0.0909 \).
- Using the approximate value \( x_0 = 1.001 \), \( x_{10} \approx 10,000,000.0909 \).

The error at \( x_{10} \) due to the small change in \( x_0 \) (from 1 to 1.001) is approximately **9,999,999.999998**, indicating a massive amplification of error through the recursive sequence.

This shows that the process is **extremely sensitive to initial conditions**, and hence **not numerically stable**.

Would you like further details or have any other questions?

Here are 5 related questions you might explore:
1. How does this instability affect the long-term behavior of the sequence?
2. Can this problem be reformulated to reduce the error propagation?
3. What does numerical stability mean in the context of recursion?
4. How would rounding errors impact the calculation of this sequence?
5. Are there other sequences or formulas where such error amplification occurs?

**Tip:** For recursive sequences, always assess the sensitivity to initial conditions, as even small changes can lead to significant differences.

对于如下序列，由 x_0 的近似值计算到 x_10 时误差有多大？计算过程是否数值稳定？

Explore the numerical stability of a recursive sequence where x_0 is slightly altered from 1 to 1.001. Learn how small changes in initial values can drastically impact the result at x_10, leading to large error propagation. Suitable for students studying numerical analysis and error propagation.

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