Problem 70 - Calculus Single Variable

Question

Calculate the values of the sequence \(a_{n}=n^{2} \cdot(1-\cos (1 / n))\) for \(1 \leq n \leq 15 .\) Now calculate \(a_{n}\) for larger values of \(n\) by setting \(n=10^{k}\) for \(1 \leq k \leq 4\). There is a number \(\ell\), which is called the limit of \(a_{n},\) such that \(\ell-a_{n}\) decreases to 0 as \(n\) increases to infinity. What is \(\ell ?\) Now plot \(f(x)=\) \((1-\cos (x)) / x^{2}\) for \(0

Step-by-Step Solution

Answer

The limit \( \ell \) of the sequence \( a_n \) is \( \frac{1}{2} \).

Step 1: Evaluate the Sequence for Small Values of n

Compute the values of the sequence for each integer in the range from 1 to 15. For instance, when \( n=1 \), \[ a_1 = 1^2 \cdot (1 - \cos(1)) \approx 1 \cdot (1 - 0.5403) = 0.4597. \]Continue calculating similarly for other values up to \( n=15 \). This will give us the values of the sequence over this range.

Step 2: Evaluate the Sequence for Larger Values of n

Compute \( a_n \) for \( n = 10^k \) where \( k = 1, 2, 3, 4 \). Calculate each using:- For \( n=10 \), use: \[ a_{10} = 10^2 \cdot (1 - \cos(0.1)) \]- For \( n=100 \), use: \[ a_{100} = 100^2 \cdot (1 - \cos(0.01)) \]- For \( n=1000 \), use: \[ a_{1000} = 1000^2 \cdot (1 - \cos(0.001)) \]- For \( n=10000 \), use: \[ a_{10000} = 10000^2 \cdot (1 - \cos(0.0001)) \]Evaluate and compare these for the limit behavior hypothesis.

Step 3: Analysis of the Limit \( \ell \)

Notice from the calculated values that as \( n \) increases, the behavior of \( a_n \) approaches a certain value. Use the Taylor series expansion for cosine: \[ \cos(x) \approx 1 - \frac{x^2}{2} + \frac{x^4}{24} - \cdots \].Plugging this into the expression for \( a_n \), as \( n \) becomes very large, \[ a_n \approx n^2 \cdot \left(1 - \left(1 - \frac{1}{2n^2}\right)\right) = \frac{1}{2}.\]Thus, the limit \( \ell \) is \( \frac{1}{2} \).

Step 4: Plot the Function f(x) and Interpret the Limit Behavior

Graph the function \( f(x) = \frac{1-\cos(x)}{x^2} \) for \( 0 < x \leq 1 \). Note that as \( x \rightarrow 0 \), \[ f(x) = \frac{1-\left(1-\frac{x^2}{2}\right)}{x^2} = \frac{x^2}{2x^2} = \frac{1}{2}. \]This graph shows that the sequence \( a_n \) approaches \( \frac{1}{2} \) as \( n \to \infty \), confirming that the limiting behavior aligns with our previous approximation for \( \ell \).

Key Concepts

Sequence evaluation

To evaluate a sequence effectively, you need to understand its behavior across different ranges of its components, especially as the variables grow larger. In our case, we evaluate the sequence \( a_n = n^2 \cdot (1 - \cos (1/n)) \) for values of \( n \) from 1 to 15. Calculating these provides insight into how \( a_n \) behaves at smaller \( n \). For example, for \( n = 1 \), the calculation gives 0.4597. You continue this for each \( n \) up to 15.

Yet, to understand the sequence more deeply, it's important to evaluate it at much larger scales of \( n \), such as \( n = 10^k \) for \( k = 1,2,3,4 \). This involves recalculating \( a_n \) at these larger values (like \( n = 10, 100, 1000, 10000 \)). Such evaluations reveal how \( a_n \) behaves and helps in hypothesizing its limit, \( \ell \). The behavior gradually settles towards a particular value which indicates the sequence's limit as \( n \) approaches infinity.

Taylor series expansion

The Taylor series is an invaluable tool in mathematics, particularly in approximating functions. By expressing \( \cos(x) \) as a Taylor series, we get: \( \cos(x) \approx 1 - \frac{x^2}{2} + \frac{x^4}{24} \cdots \). This expansion allows us to simplify and understand the limiting behavior of sequences.

In the case of our sequence \( a_n = n^2 \cdot (1 - \cos (1/n)) \), substituting the Taylor expansion, particularly \( \cos(1/n) \approx 1 - \frac{1}{2n^2} \) can significantly simplify calculation as \( n \) becomes very large. As such, \( a_n \approx n^2 \cdot \left(1 - \left(1 - \frac{1}{2n^2}\right)\right) \), simplifying to \( \frac{1}{2} \) for a very large \( n \). This demonstrates that the sequence's limit \( \ell \) is \( \frac{1}{2} \), which aligns with our hypothesis.

Graphical analysis

Graphical analysis is a powerful method to visually interpret the behavior of functions and sequences. By graphing \( f(x) = \frac{1 - \cos(x)}{x^2} \) for \( 0 < x \leq 1 \), you can observe how the function behaves as \( x \) approaches zero.

The function's value near \( x \rightarrow 0 \) simplifies to \( \frac{1}{2} \), which helps confirm the calculated limit of the sequence \( a_n \). The graph shows a stabilization towards \( \frac{1}{2} \), further solidifying our understanding that as \( n \rightarrow \infty \), \( a_n \) does indeed approach its limit \( \ell \) of \( \frac{1}{2} \).

Such visual aids are critical in comprehending complex mathematical relationships at a glance, enhancing our comprehension of theoretical and practical aspects of sequences.