The given infinite series is:

$1 + \frac{1}{4} + \frac{1}{9} + \frac{1}{16} + \frac{1}{25} + \dots$

This is the sum of the reciprocals of the squares of the positive integers, which is a well-known infinite series:

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

The sum of this series is known as the Basel problem, and the solution is:

$\sum_{n=1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6}$

Now, let's approximate $\frac{\pi^2}{6}$.

$\pi^2 \approx 9.8696$

$\frac{\pi^2}{6} \approx \frac{9.8696}{6} \approx 1.64493$

Thus, the integer part of the sum is:

$\boxed{1}$

Would you like more details or have any questions?

Here are 5 related questions to explore:

1. How is the Basel problem related to the Riemann zeta function?
2. What is the sum of the series $\sum_{n=1}^{\infty} \frac{1}{n^4}$?
3. How can we estimate the sum of a convergent series numerically?
4. What is the relationship between infinite series and improper integrals?
5. What are some applications of infinite series in physics and engineering?

Tip: To recognize the sum of an infinite series, consider looking for known patterns like geometric, harmonic, or power series.