Sure! Here’s a very challenging mathematics problem that requires a deep understanding of calculus, series, and mathematical rigor:

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The Basel Problem (with a twist):

Evaluate the following infinite series:

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

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Hints:

1. This is a modified version of the classic Basel problem (which deals with the sum of $\sum \frac{1}{n^2}$).
2. You need to manipulate the general term to find a closed form.
3. Consider partial fraction decomposition or transformations involving generating functions.
4. It might also help to rewrite the term as: $\frac{1}{n^2 + n + 1} = \frac{1}{(n+1)(n+2) - 1}$

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This problem blends techniques from infinite series, partial fractions, and calculus. It will likely take multiple steps and require persistence to crack!

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Would you like hints to proceed or additional details on techniques you could apply?

5 Related Questions:

1. What is the sum of the series $\sum_{n=1}^{\infty} \frac{1}{n^2}$?
2. How can we use partial fractions to solve series involving quadratic terms?
3. What are the convergence criteria for an infinite series?
4. How does Euler’s work on the Basel problem apply to modern analysis?
5. Can we use generating functions to find closed forms for such series?

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Tip:
When stuck with an infinite series, look for telescoping patterns or relate it to known series (like harmonic or geometric series).