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The task asks to calculate the value of the given series and find its limit:

\[
\sum_{k=1}^{\infty} \frac{2}{k^2 + k}
\]

### Solution:

We begin by simplifying the term inside the sum. The general term can be rewritten as follows:

\[
\frac{2}{k^2 + k} = \frac{2}{k(k+1)}
\]

Now, using partial fraction decomposition, we can break this down:

\[
\frac{2}{k(k+1)} = \frac{A}{k} + \frac{B}{k+1}
\]

Multiplying both sides by \( k(k+1) \) gives:

\[
2 = A(k+1) + Bk
\]

Expanding the right-hand side:

\[
2 = A k + A + B k
\]

This implies:

\[
A + B = 0 \quad \text{and} \quad A = 2
\]

From \( A = 2 \), we know \( B = -2 \). Therefore, the partial fraction decomposition is:

\[
\frac{2}{k(k+1)} = \frac{2}{k} - \frac{2}{k+1}
\]

### The Infinite Series

Now, substituting this back into the original sum:

\[
\sum_{k=1}^{\infty} \left( \frac{2}{k} - \frac{2}{k+1} \right)
\]

This is a telescoping series. In a telescoping series, most terms cancel out, leaving only a few terms from the beginning and the end. Writing out the first few terms:

\[
\left( \frac{2}{1} - \frac{2}{2} \right) + \left( \frac{2}{2} - \frac{2}{3} \right) + \left( \frac{2}{3} - \frac{2}{4} \right) + \dots
\]

Most terms cancel, and we are left with:

\[
\lim_{n \to \infty} \left( 2 - \frac{2}{n+1} \right)
\]

As \( n \to \infty \), \( \frac{2}{n+1} \to 0 \). Therefore, the sum of the series is:

\[
2
\]

### Conclusion:

The series converges to **2**.

---

Would you like more details or have any questions about this solution?

Here are 5 related questions to deepen your understanding:
1. What are the properties of a telescoping series?
2. How does partial fraction decomposition work in other cases?
3. Can you generalize this method for other series involving rational terms?
4. How do you determine whether a series converges?
5. What is the importance of the limit in infinite series?

**Tip:** In telescoping series, always check which terms cancel to simplify your sum.

Berechnen Sie den Reihenwert der gegebenen Reihe: Summe von 2/(k^2+k) von k=1 bis unendlich. Stellen Sie hierfür die Partialsumme sm auf und bilden Sie den Grenzwert lim m->∞ sm.

Learn how to calculate the sum of the infinite series 2/(k^2+k) using the telescoping series method and partial fraction decomposition. This detailed explanation is suitable for students in advanced high school or early university calculus courses.

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