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

Step 1: Understanding the Concept:
The convergence of an infinite series can be determined using several tests.
Two of the most common tests are D'Alembert's Ratio Test and Cauchy's Root Test.

Step 3: Detailed Explanation:
Let us analyze D'Alembert's Ratio Test for an infinite series \(\sum a_n\) of non-zero real numbers.
We calculate the limit of the absolute ratio of consecutive terms:
\[ L = \lim_{n\to\infty} \left|\frac{a_{n+1}}{a_n}\right| \]
The test states that:
1. If \(L < 1\), the series \(\sum a_n\) is absolutely convergent (and therefore convergent).
2. If \(L > 1\) (or \(L = \infty\)), the series is divergent.
3. If \(L = 1\), the test is inconclusive, and the series may converge or diverge.
In competitive exam questions, the condition \(L < 1\) is often written as the primary requirement for convergence under this test.
Let us evaluate the options:
- Option (C) represents this standard ratio test condition for convergence:
\[ \lim_{n\to\infty} \left|\frac{a_{n+1}}{a_n}\right| < 1 \]
- Options (A) and (B) describe Cauchy's Root Test, but they state \(> 1\) (which causes divergence) or \(= 1\) (which is inconclusive).
Thus, Option (C) is the correct statement.

Step 4: Final Answer:
Therefore, the correct option is (C).