How do you determine whether the sequence #a_n=(1)^n/sqrtn# converges, if so how do you find the limit?
Limit is 0
Now, use the theorem of comparison to conclude.
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To determine whether the sequence (a_n = \frac{(1)^n}{\sqrt{n}}) converges, you can analyze its behavior as (n) approaches infinity.
Firstly, note that the sequence oscillates between positive and negative values as (n) increases.
Secondly, consider the behavior of (\frac{1}{\sqrt{n}}) as (n) approaches infinity. As (n) grows larger, the denominator (\sqrt{n}) also grows, which causes the fraction (\frac{1}{\sqrt{n}}) to approach zero.
However, because the sequence (a_n) alternates between positive and negative values, it does not converge to a single limit.
Therefore, the sequence (a_n = \frac{(1)^n}{\sqrt{n}}) does not converge.
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To determine whether the sequence (a_n = \frac{(1)^n}{\sqrt{n}}) converges, we can analyze its behavior as (n) approaches infinity.
Since (n) is approaching infinity, the oscillating term ((1)^n) alternates between (1) and (1) as (n) increases. However, the denominator (\sqrt{n}) grows without bound as (n) increases.
For even values of (n), (a_n = \frac{1}{\sqrt{n}}), and for odd values of (n), (a_n = \frac{1}{\sqrt{n}}). As (n) approaches infinity, both (\frac{1}{\sqrt{n}}) and (\frac{1}{\sqrt{n}}) tend towards (0).
Therefore, the sequence (a_n) converges to (0) as (n) approaches infinity.
The limit of the sequence (a_n) as (n) approaches infinity is (0).
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To determine whether the sequence (a_n = \frac{(1)^n}{\sqrt{n}}) converges, we can analyze its behavior as (n) approaches infinity.
The sequence alternates between positive and negative terms as (n) changes. However, the terms (1/\sqrt{n}) decrease as (n) increases, approaching zero.
To find the limit, we need to consider two cases: when (n) is even and when (n) is odd.

When (n) is even, (a_n = \frac{1}{\sqrt{n}}). As (n) approaches infinity, (1/\sqrt{n}) approaches zero.

When (n) is odd, (a_n = \frac{1}{\sqrt{n}}). As (n) approaches infinity, (1/\sqrt{n}) also approaches zero.
Since both cases converge to zero, the sequence (a_n) converges to zero as (n) approaches infinity.
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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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