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AlekseyPX
3 years ago
14

Use the Ratio Test to determine whether the series is convergent or divergent. [infinity] n! 112n n = 1 Identify an. Correct: Yo

ur answer is correct. Evaluate the following limit. lim n → [infinity] an + 1 an Since lim n → [infinity] an + 1 an 1,
Mathematics
1 answer:
MakcuM [25]3 years ago
7 0

Answer:

Step-by-step explanation:

Recall that the ratio test is stated as follows:

Given a series of the form \sum_{n=1}^{\infty} a_n let L=\lim_{n\to \infty}\left|\frac{a_{n+1}}{a_n}\right|

If L<1, then the series converge absolutely, if L>1, then the series diverge. If L fails to exist or L=1, then the test is inconclusive.

Consider the given series \sum_{n=1}^{\infty} n! \cdot 112n. In this case, a_n =n! \cdot 112n, so , consider the limit

\lim_{n\to\infty} \frac{(n+1)! 112 (n+1)}{n! 112 n} = \lim_{n\to\infty}\frac{(n+1)^2}{n}

Since the numerator has a greater exponent than the numerator, the limit is infinity, which is greater than one, hence, the series diverge by the ratio test

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