How do you use the limit comparison test on the series #sum_(n=1)^oo(n^25n)/(n^3+n+1)# ?
By using the leading terms of the numerator and the denominator, we can construct
By Limit Comparison Test,
#lim_{n to infty}{a_n}/{b_n}=lim_{n to infty}{n^25n}/{n^3+n+1}cdot n/1 =lim_{n to infty}{n^35n^2}/{n^3+n+1}#
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To use the limit comparison test on the series (\sum_{n=1}^{\infty} \frac{n^2  5n}{n^3 + n + 1}), follow these steps:

Choose a comparison series (b_n) that is positive for all (n) and whose convergence or divergence is known.

Take the limit of the ratio of the given series and the comparison series as (n) approaches infinity: [ \lim_{n \to \infty} \frac{a_n}{b_n} ]

If the limit is a finite positive number, both series either converge or diverge. If it's zero, the given series converges; if it's infinity, the given series diverges.
Choose a comparison series that simplifies the expression. For this series, let's use (b_n = \frac{1}{n}), a pseries with (p = 1), which is known to diverge.
Now, compute the limit: [ \lim_{n \to \infty} \frac{\frac{n^2  5n}{n^3 + n + 1}}{\frac{1}{n}} ] [ = \lim_{n \to \infty} \frac{n^2  5n}{n^3 + n + 1} \cdot n ] [ = \lim_{n \to \infty} \frac{n^3  5n^2}{n^3 + n + 1} ]
Now, divide each term by the highest power of (n) in the denominator: [ = \lim_{n \to \infty} \frac{1  \frac{5}{n}}{1 + \frac{1}{n^2} + \frac{1}{n^3}} ]
As (n) approaches infinity, the terms with (1/n) approach zero: [ = \frac{1}{1} = 1 ]
Since the limit is a finite positive number, the given series and the comparison series have the same behavior. Therefore, both series diverge.
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When evaluating a onesided 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 onesided 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 onesided 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 onesided 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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