Question

The series $\sum_{n=1}^{\infty} \frac{x^n}{n+1}$ is :

• A. divergent for all x
• B. convergent if $x < 1$ and divergent if $x > 1$
• C. convergent if $x > 1$ and divergent if $x < 1$
• D. convergent for all x

Hint:

For any standard power series of the form $\sum \frac{x^n}{n^p}$, the radius of convergence $R$ is always equal to 1, meaning convergence is bounded by the interval $(-1, 1)$.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
The convergence of a power series can be analyzed by determining its radius of convergence using the D'Alembert's Ratio Test.
Key Formula or Approach:
For a series $\sum u_n(x)$ with positive terms, the series converges if:
\[ \lim_{n\to\infty} \left| \frac{u_{n+1}}{u_n} \right| < 1 \]

Step 2: Detailed Explanation:
Let the general term of the series be $u_n = \frac{x^n}{n+1}$.
Apply the Ratio Test:
\[ \lim_{n\to\infty} \left| \frac{u_{n+1}}{u_n} \right| = \lim_{n\to\infty} \left| \frac{x^{n+1}}{n+2} \cdot \frac{n+1}{x^n} \right| \]
\[ = |x| \lim_{n\to\infty} \frac{n+1}{n+2} = |x| \cdot 1 = |x| \]
According to the Ratio Test:
- The series converges if $|x| < 1$, which corresponds to $-1 < x < 1$.
- The series diverges if $|x| > 1$.
For positive values of $x$, this means:
- The series is convergent if $x < 1$.
- The series is divergent if $x > 1$.
Therefore, the correct choice is convergent if $x < 1$ and divergent if $x > 1$.

Step 3: Final Answer
The correct option is (B).