Series Analysis for Divergence

The question requires identifying the series that diverges from the given options. We analyze each series using standard convergence tests.

Option 1 Analysis: Geometric Series

Series: $\sum_{n=1}^{\infty} \frac{1}{3^{n-1}}$

• This is a geometric series with first term $a = \frac{1}{3^{1-1}} = 1$ and common ratio $r = \frac{1}{3}$.
• A geometric series converges if the absolute value of the common ratio is less than 1 ($|r| < 1$).
• Here, $|r| = |\frac{1}{3}| < 1$.
• Conclusion: This series converges.

Option 2 Analysis: Harmonic Series

Series: $\sum_{n=1}^{\infty} \frac{1}{n}$

• This is the harmonic series.
• Conclusion: The harmonic series is a well-known divergent series.

Option 3 Analysis: Geometric Series

Series: $\sum_{n=0}^{\infty} \frac{1}{2^n}$

• This is a geometric series with first term $a = \frac{1}{2^0} = 1$ and common ratio $r = \frac{1}{2}$.
• A geometric series converges if $|r| < 1$.
• Here, $|r| = |\frac{1}{2}| < 1$.
• Conclusion: This series converges.

Option 4 Analysis: Root Test

Series: $\sum_{n=1}^{\infty} \frac{1}{n^n}$

• We apply the Root Test: $L = \lim_{n \to \infty} \sqrt[n]{|a_n|}$.
• $a_n = \frac{1}{n^n}$.
• Calculate the limit: $ L = \lim_{n \to \infty} \left| \frac{1}{n^n} \right|^{1/n} = \lim_{n \to \infty} \frac{1}{n} $
• $L = 0$.
• The Root Test states that if $L < 1$, the series converges.
• Conclusion: This series converges.

Final Identification of Divergent Series

Comparing the conclusions for each option:

• Option 1: Converges
• Option 2: Diverges
• Option 3: Converges
• Option 4: Converges

The only divergent series is the harmonic series found in Option 2.