Question

Let \(\sum_{n=1}^\infty a_n\) be series of real number, then:

• A. \(\sum_{n=1}^\infty a_n\) is convergent if and only if \(\lim_{n\to\infty} |a_n|^{\frac{1}{n}} > 1\)
• B. \(\sum_{n=1}^\infty a_n\) is convergent if and only if \(\lim_{n\to\infty} |a_n|^{\frac{1}{n}} = 1\)
• C. \(\sum_{n=1}^\infty a_n\) is convergent if and only if \(\lim_{n\to\infty} \left|\frac{a_{n+1}}{a_n}\right| < 1\)
• D. \(\sum_{n=1}^\infty a_n\) is convergent if and only if \(\lim_{n\to\infty} \left|\frac{a_{n+1}}{a_n}\right| = 1\)

Hint:

Remember the conditions for both the Ratio and Root tests:
- For convergence, the limit must be strictly less than 1 (\(L < 1\)).
- If \(L = 1\), the test fails, and you must use a different test (like Raabe's or Gauss's test).

Answer 1

The Correct Option is C