How do you know if the summation #(-12)^n/n# where n is between 3 to infinity is convergent or divergent?

Answer 1
#sum_(n=3)^oo(-12)^n/n# is divergent.
You can determine that this series diverges because the terms do not go to #0# as #n# increases without bound. That is: #lim_(nrarroo)((-12)^n)/n!=0#.
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Answer 2
To determine if the series \( \sum_{n=3}^{\infty} \frac{(-12)^n}{n} \) is convergent or divergent, we can use the ratio test. The ratio test states that if \( \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| < 1 \), then the series \( \sum_{n=1}^{\infty} a_n \) converges. If \( \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| > 1 \) or the limit does not exist, then the series diverges. If \( \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = 1 \), the test is inconclusive. For the series \( \sum_{n=3}^{\infty} \frac{(-12)^n}{n} \), let's calculate the ratio: \[ \left| \frac{a_{n+1}}{a_n} \right| = \left| \frac{\frac{(-12)^{n+1}}{n+1}}{\frac{(-12)^n}{n}} \right| \] \[ = \left| \frac{(-12)^{n+1} \cdot n}{(-12)^n \cdot (n+1)} \right| \] \[ = \left| \frac{(-12)(n)}{n+1} \right| \] \[ = \left| \frac{-12n}{n+1} \right| \] Now, as \( n \) approaches infinity, the absolute value of this expression tends towards \( 12 \). Since \( 12 > 1 \), according to the ratio test, the series diverges. Therefore, the series \( \sum_{n=3}^{\infty} \frac{(-12)^n}{n} \) is divergent.
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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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