Find the limit of the sequence whose general term is a sub n = 1 - 3n/ 4 + n. (5 points) I got Does not Exist. But I think it is wrong. Am I doing this right?
A) 1/4
B) 3
C) −3
D) Does not exist
You need some parentheses
Clearly, as written, the sequence diverges, since it is just 1 + n/4
But
(1-3n)/(4+n) → -3n/n = -3
To find the limit of the sequence, we can take the limit as n approaches infinity of the general term a sub n.
The general term is given by a sub n = 1 - 3n/4 + n.
Taking the limit as n approaches infinity means we need to analyze the behavior of the sequence as n gets larger and larger.
The limit of a sequence exists if and only if it approaches a finite number as n approaches infinity. If the terms of the sequence become arbitrarily large or the sequence oscillates without settling to a single value, then the limit does not exist.
Let's simplify the general term:
a sub n = 1 - 3n/4 + n
= 1 + n - 3n/4
As n approaches infinity, the term "n" dominates while the term "-3n/4" becomes insignificant.
So, taking the limit as n approaches infinity:
lim (n -> ∞) (1 + n - 3n/4)
= ∞ (since n dominates)
Therefore, the limit of the sequence does not exist (option D - Does not exist) since it does not approach a finite number.
So, your initial answer of "Does not exist" was correct.
To find the limit of the given sequence, you can start by looking at the general term:
a sub n = 1 - (3n/4) + n
Next, let's find the expression for the limit as n approaches infinity:
lim(n→∞) a sub n
To find the limit, you can observe the behavior of the sequence as n becomes larger and larger. We can simplify the expression by grouping the terms:
lim(n→∞) (1 - (3n/4) + n)
Now, we can analyze each term separately. The first term, 1, does not depend on n, so it remains constant as n approaches infinity.
For the second term, -(3n/4), as n becomes larger, the absolute value of this term grows indefinitely. However, the negative sign causes it to tend towards negative infinity.
Finally, for the third term, n, as n approaches infinity, this term also grows indefinitely.
As a result, when we combine these terms, we have a sum of a constant term, a term going towards negative infinity, and a term going towards positive infinity. Since the positive and negative infinities do not cancel out, the limit of the sequence does not exist.
Therefore, the correct answer is:
D) Does not exist