We are given the series:

$s = \sum_{n=1}^{\infty} \frac{1}{n^2 + 9}$

and we are asked to bound the error of the approximation of $s$ by the partial sum $s_{20}$, which is:

$s_{20} = \sum_{n=1}^{20} \frac{1}{n^2 + 9}$

The Integral Test can be used to estimate the error when truncating the series after a certain number of terms. According to the Integral Test, the error $|s - s_{20}|$ is bounded by the integral:

$\int_{N+1}^{\infty} f(x) \, dx$

where $f(x) = \frac{1}{x^2 + 9}$ is the function corresponding to the terms of the series, and $N = 20$ in this case. We need to compute the integral from $21$ to infinity to find an upper bound for the error.

Step 1: Find the integral

We need to compute:

$\int_{21}^{\infty} \frac{1}{x^2 + 9} \, dx$

This is a standard integral, and its solution can be found using the arctangent formula:

$\int \frac{1}{x^2 + a^2} \, dx = \frac{1}{a} \arctan\left( \frac{x}{a} \right)$

In our case, $a = 3$, so we get:

$\int_{21}^{\infty} \frac{1}{x^2 + 9} \, dx = \left[ \frac{1}{3} \arctan\left( \frac{x}{3} \right) \right]_{21}^{\infty}$

Step 2: Evaluate the integral

At the upper limit, as $x \to \infty$, we have:

$\arctan\left( \frac{\infty}{3} \right) = \frac{\pi}{2}$

At the lower limit, $x = 21$:

$\arctan\left( \frac{21}{3} \right) = \arctan(7)$

Thus, the integral becomes:

$\frac{1}{3} \left( \frac{\pi}{2} - \arctan(7) \right)$

We can now calculate the value of $\arctan(7)$. Using a calculator:

$\arctan(7) \approx 1.4289 \, \text{radians}$

Substituting this into the expression:

$\frac{1}{3} \left( \frac{\pi}{2} - 1.4289 \right) = \frac{1}{3} \left( 1.5708 - 1.4289 \right) = \frac{1}{3} \times 0.1419 \approx 0.0473$

Step 3: Conclusion

Thus, the upper bound for the error is approximately:

$|s - s_{20}| \leq 0.0473$

Therefore, the upper bound for the error, accurate to three significant digits, is:

$\boxed{0.0473}$

Do you need any more details or explanations?

Here are some related questions:

1. How can we calculate the error bound for a different value of $N$?
2. What is the role of the arctangent function in the Integral Test?
3. How would the error bound change if we truncated the series after a larger number of terms?
4. What is the importance of the Integral Test in estimating the error in infinite series?
5. How can we use the error bound to check the accuracy of an approximation for $s$?

Tip: The Integral Test provides a way to estimate the error of a truncated series when the series terms decrease steadily.