Here is another convergent infinite series

1+1/4+1/9+1/16+1/25+...

What is the integer part of its sum? 

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.

Here is another convergent infinite series

1+1/4+1/9+1/16+1/25+...

What is the integer part of its sum?

Explore the solution to the convergent infinite series 1 + 1/4 + 1/9 + ... and its connection to the Basel Problem. Learn how the sum of this series equals π²/6 and how to find the integer part of the result. Suitable for Grades 11-12 and early college students.

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