# How do you use the limit comparison test to determine if #Sigma tan(1/n)# from #[1,oo)# is convergent or divergent?

Let us check the hypotheses of Limit Comparison Theorem.

By l'Hospital's Rule,

#=lim_(n to infty)(sec^2(1/n)cdot cancel(-1/n^2)) /cancel(-1/n^2)=sec^2(0)=1#

I hope that this was clear.

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To determine the convergence of the series ( \sum \tan\left(\frac{1}{n}\right) ) from ( n = 1 ) to ( n = \infty ) using the Limit Comparison Test:

- Choose a series ( \sum b_n ) that you know the convergence of.
- Compute the limit ( \lim_{{n \to \infty}} \frac{\tan\left(\frac{1}{n}\right)}{b_n} ).
- If the limit is a positive finite number, then both series ( \sum \tan\left(\frac{1}{n}\right) ) and ( \sum b_n ) either both converge or both diverge.

Let's choose ( b_n = \frac{1}{n} ), a series we know to be divergent (it's the harmonic series).

Now, we calculate the limit:

[ \lim_{{n \to \infty}} \frac{\tan\left(\frac{1}{n}\right)}{\frac{1}{n}} ]

Using the limit ( \lim_{{x \to 0}} \frac{\tan(x)}{x} = 1 ), which is a well-known limit, we simplify:

[ \lim_{{n \to \infty}} n \tan\left(\frac{1}{n}\right) = \lim_{{n \to \infty}} \frac{\tan\left(\frac{1}{n}\right)}{\frac{1}{n}} = 1 ]

Since this limit is a positive finite number (1), by the Limit Comparison Test, ( \sum \tan\left(\frac{1}{n}\right) ) and ( \sum \frac{1}{n} ) either both converge or both diverge.

Since we know that ( \sum \frac{1}{n} ) diverges (harmonic series), we conclude that ( \sum \tan\left(\frac{1}{n}\right) ) also diverges.

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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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