Problem 38
Question
\(37-40=\) Determine whether the sequence is increasing, decreasing, or not monotonic. Is the sequence bounded? $$a_{n}=\frac{2 n-3}{3 n+4}$$
Step-by-Step Solution
Verified Answer
The sequence is not monotonic and is bounded.
1Step 1: Determine the nature of the sequence
To check if the sequence \(a_n = \frac{2n - 3}{3n + 4}\) is increasing or decreasing, find the difference \(a_{n+1} - a_n\). Calculate \(a_{n+1} = \frac{2(n+1) - 3}{3(n+1) + 4} = \frac{2n - 1}{3n + 7}\). Compare \(a_{n+1}\) and \(a_n\).
2Step 2: Calculate the difference
Subtract \(a_n\) from \(a_{n+1}\):\[a_{n+1} - a_n = \frac{2n - 1}{3n + 7} - \frac{2n - 3}{3n + 4} = \frac{(2n - 1)(3n + 4) - (2n - 3)(3n + 7)}{(3n + 7)(3n + 4)}\]Simplify the numerator.
3Step 3: Simplify the numerator and analyze
Simplify:\[(2n - 1)(3n + 4) - (2n - 3)(3n + 7) = (6n^2 + 8n - 3n - 4) - (6n^2 + 14n - 9n - 21)\]\[= 5n + 17 - 5n - 21\]\[= 0\] The result is \(0\), meaning \(a_{n+1} = a_n\), indicating the sequence is not monotonic.
4Step 4: Check if the sequence is bounded
Observe that as \(n\) approaches infinity, \(a_n = \frac{2n - 3}{3n + 4}\) approaches \(\frac{2}{3}\). The function is bounded as it approaches a single value without going to infinity. Similarly, for \(n = 1\), the sequence value is positive. Thus, the sequence is bounded between some finite values.
Key Concepts
Monotonic SequencesBounded SequencesSequence Convergence
Monotonic Sequences
In mathematics, a monotonic sequence is one that is consistently either non-decreasing or non-increasing. A sequence is increasing if each element is greater than or equal to the previous one, whereas it is said to be decreasing if each element is less than or equal to the previous one. A sequence can be perfectly monotonic, which means all its parts are entirely increasing or decreasing.
To determine if a sequence is monotonic, one method is to examine the difference between consecutive terms. For a sequence defined as \(a_n = \frac{2n - 3}{3n + 4}\), we compute the difference between \(a_{n+1}\) and \(a_n\). In our case, this difference simplifies to zero, indicating that the sequence neither increases nor decreases uniformly. Thus, it is not considered monotonic because there is no consistent pattern in either direction.
Monotonicity is crucial because it helps us understand the behavior of sequences, particularly when analyzing convergence or applying differential calculus.
To determine if a sequence is monotonic, one method is to examine the difference between consecutive terms. For a sequence defined as \(a_n = \frac{2n - 3}{3n + 4}\), we compute the difference between \(a_{n+1}\) and \(a_n\). In our case, this difference simplifies to zero, indicating that the sequence neither increases nor decreases uniformly. Thus, it is not considered monotonic because there is no consistent pattern in either direction.
Monotonicity is crucial because it helps us understand the behavior of sequences, particularly when analyzing convergence or applying differential calculus.
Bounded Sequences
A bounded sequence is a sequence where all values lie within a specific range, without reaching positive or negative infinity. This means that there are certain upper and lower limits that the sequence never exceeds.
For instance, consider the sequence \(a_n = \frac{2n - 3}{3n + 4}\). As \(n\) approaches infinity, it approaches \(\frac{2}{3}\). This means the sequence is bounded above by values approaching \(\frac{2}{3}\). Additionally, checking for various initial values of \(n\) shows that the sequence remains positive. Thus, it is also bounded below by zero.
Bounding sequences are particularly significant because they guarantee that a sequence will avoid diverging to infinity, making it possible to analyze its behavior over an extended range. Boundedness often also contributes to the sequence's convergence.
For instance, consider the sequence \(a_n = \frac{2n - 3}{3n + 4}\). As \(n\) approaches infinity, it approaches \(\frac{2}{3}\). This means the sequence is bounded above by values approaching \(\frac{2}{3}\). Additionally, checking for various initial values of \(n\) shows that the sequence remains positive. Thus, it is also bounded below by zero.
Bounding sequences are particularly significant because they guarantee that a sequence will avoid diverging to infinity, making it possible to analyze its behavior over an extended range. Boundedness often also contributes to the sequence's convergence.
Sequence Convergence
Sequence convergence is an essential concept indicating that as the index \(n\) becomes extremely large, the terms \(a_n\) approach a specific value, known as the limit. For a sequence to converge, this limit must be finite, and the sequence must approach it steadily without oscillations.
In the case of the sequence \(a_n = \frac{2n - 3}{3n + 4}\), it is shown that as \(n\) grows indefinitely, the sequence approaches the value \(\frac{2}{3}\). To prove convergence formally, one would demonstrate that for each small positive number \(\epsilon\), there exists a point in the sequence beyond which all subsequent terms are within \(\epsilon\) of \(\frac{2}{3}\).
Convergence is a powerful property that allows mathematicians and scientists to predict long-term behavior of sequences, making them easier to analyze and apply in mathematical models.
In the case of the sequence \(a_n = \frac{2n - 3}{3n + 4}\), it is shown that as \(n\) grows indefinitely, the sequence approaches the value \(\frac{2}{3}\). To prove convergence formally, one would demonstrate that for each small positive number \(\epsilon\), there exists a point in the sequence beyond which all subsequent terms are within \(\epsilon\) of \(\frac{2}{3}\).
Convergence is a powerful property that allows mathematicians and scientists to predict long-term behavior of sequences, making them easier to analyze and apply in mathematical models.
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