Does the sequence converge or diverge?
2n+(-1)^n5/ 4n-(-1)^n3
I know I need to find the limit..but i don't know how to do this
To determine whether the sequence converges or diverges, we need to find the limit of the sequence as n approaches infinity. Let's break down the steps to find the limit:
1. Rewrite the given expression: 2n + (-1)^n5 / 4n - (-1)^n3
2. Simplify the numerator and the denominator separately.
For the numerator, we have 2n + (-1)^n5. The term (-1)^n alternates between -1 and 1 as n increases. Since we are considering n approaching infinity, we can see that the term (-1)^n5 will flip between -5 and 5. So, we can rewrite the numerator as: 2n + (-5) when n is odd and 2n + 5 when n is even.
For the denominator, we have 4n - (-1)^n3. Similar to the numerator, (-1)^n alternates between -1 and 1. This time, the term (-1)^n3 will flip between -3 and 3. Thus, we can rewrite the denominator as: 4n + 3 when n is odd and 4n - 3 when n is even.
3. Combine the numerator and denominator.
Now we have established that when n is odd, the expression becomes (2n - 5) / (4n + 3), and when n is even, the expression becomes (2n + 5) / (4n - 3).
4. Divide both the numerator and denominator by n.
Dividing both the numerator and denominator by n allows us to focus on the leading terms of the expression when n approaches infinity. This step helps eliminate the lower-order terms that do not have a significant effect on the overall limit. After dividing, we get:
When n is odd: (2 - 5/n) / (4 + 3/n)
When n is even: (2 + 5/n) / (4 - 3/n)
5. Evaluate the limit.
To find the limit, we evaluate the term when n approaches infinity. Let's consider the case when n is odd:
The limit as n approaches infinity of (2 - 5/n) / (4 + 3/n) can be found by examining the dominant terms as n becomes larger. The leading terms are 2/n in the numerator and 4/n in the denominator. Since both terms have a coefficient of 1/n, the limit of the expression is (2/n) / (4/n) = 2/4 = 1/2.
Similarly, when n is even, we find that the limit of (2 + 5/n) / (4 - 3/n) is also 1/2.
6. Conclude whether the sequence converges or diverges.
Since the limit of the expression is the same regardless of whether n is odd or even, we can conclude that the given sequence converges. The limit is equal to 1/2.
Note: It is essential to carefully analyze the numerator and denominator's behavior to determine if any terms dominate the expression as n approaches infinity. Dividing both the numerator and denominator by n is a common technique to simplify the expression and focus on the leading terms.