1 Aufgabe 37 Potenzreihen Entscheiden Sie fr jedes x R ob die Potenzreihe n 1 xnn konvergiert bzw absolut konvergiert

**Solution By Steps** *Step 1:* Determine the interval of convergence using the ratio test. The ratio test states that for a series $\sum a_n$, if $\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = L$ and: - If $L < 1$, the series converges absolutely. - If $L > 1$, the series diverges. - If $L = 1$, the test is inconclusive. For the given series $\sum_{n=1}^{\infty} \frac{x^{n}}{n}$, let's denote $a_n = \frac{x^n}{n}$. We calculate the limit: $$ L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = \lim_{n \to \infty} \left| \frac{x^{n+1}/(n+1)}{x^n/n} \right| = \lim_{n \to \infty} \left| \frac{x \cdot n}{n+1} \right| = |x|. $$ *Step 2:* Apply the result of the ratio test. Since $L = |x|$, we have: - If $|x| < 1$, the series converges absolutely. - If $|x| > 1$, the series diverges. - If $|x| = 1$, the test is inconclusive, and we need to check the endpoints $x = 1$ and $x = -1$ separately. *Step 3:* Check the convergence at the endpoints $x = 1$ and $x = -1$. For $x = 1$, the series becomes $\sum_{n=1}^{\infty} \frac{1}{n}$, which is the harmonic series and is known to diverge. For $x = -1$, the series becomes $\sum_{n=1}^{\infty} \frac{(-1)^n}{n}$, which is the alternating harmonic series and is known to converge conditionally by the alternating series test. **Final Answer** The series $\sum_{n=1}^{\infty} \frac{x^{n}}{n}$ converges absolutely for $|x| < 1$, converges conditionally for $x = -1$, and diverges for $x = 1$ and $|x| > 1$. **Key Concept** Convergence Tests **Key Concept Explanation** Convergence tests are methods used to determine whether an infinite series converges or diverges. The ratio test, in particular, compares the limit of the absolute ratio of successive terms. If this limit is less than one, the series converges absolutely; if more than one, it diverges; and if equal to one, the test is inconclusive, requiring further investigation. Absolute convergence implies that the series converges even when all terms are replaced by their absolute values, while conditional convergence means the series converges but does not converge absolutely.