Determine whether the given series is convergent.
${\sum }_{n=1}^{\infty }{n}^{-1}+n^{-3}$

• A. Diverges
• B. Converges
• C. Both converges and diverges

Answer: (A)

Verified step by step guidance

1. Step 1: Understand the problem. The series given is \( \sum_{n=1}^{\infty} n^{-1} + n^{-3} \). We need to determine whether this series converges or diverges. To do this, analyze each term separately: \( n^{-1} \) and \( n^{-3} \).
2. Step 2: Recall the p-series test. A p-series is of the form \( \sum_{n=1}^{\infty} \frac{1}{n^p} \), and it converges if \( p > 1 \) and diverges if \( p \leq 1 \). Apply this test to each term in the series.
3. Step 3: Analyze \( n^{-1} \). Here, \( p = 1 \). Since \( p \leq 1 \), the series \( \sum_{n=1}^{\infty} n^{-1} \) diverges. This is known as the harmonic series, which is a classic example of divergence.
4. Step 4: Analyze \( n^{-3} \). Here, \( p = 3 \). Since \( p > 1 \), the series \( \sum_{n=1}^{\infty} n^{-3} \) converges. This term alone would contribute to convergence.
5. Step 5: Combine the results. The series \( \sum_{n=1}^{\infty} n^{-1} + n^{-3} \) is the sum of a divergent series (\( n^{-1} \)) and a convergent series (\( n^{-3} \)). Since the divergence of \( n^{-1} \) dominates, the entire series \( \sum_{n=1}^{\infty} n^{-1} + n^{-3} \) diverges.