Question

Do the following: a. Compute: \(\lim _{n \rightarrow \infty} n \ln \left(1-\frac{3}{n}\right)\). b. Use L'Hopital's Rule to evaluate \(L=\lim _{x \rightarrow \infty}\left(1-\frac{4}{x}\right)^{x}\). [Hint: Consider \(\ln L\).] c. Determine the convergence of \(\sum_{n=1}^{\infty}\left(\frac{n}{3 n+2}\right)^{n^{2}}\). d. Sum the series \(\sum_{n=1}^{\infty}\left[\tan ^{-1} n-\tan ^{-1}(n+1)\right]\) by first writing the \(N\) th partial sum and then computing \(\lim _{N \rightarrow \infty} s_{N}\).

Short answer

a. The limit is -3. b. The limit of L is \(e^{-4}\). c. The series diverges. d. The sum of the series is \(\pi/4\).

Step-by-step solution

Solve Problem a

Recognising the limit is in the indeterminate form \(-\infty \cdot 0\), rewrite the limit as a quotient of two functions, \( -\frac{\ln\left(1-\frac{3}{n}\right)}{1/n}\). Now, the limit appears in the `0/0` indeterminate form and can be evaluated using L'Hopital's rule. Applying L'Hopital's rule repeatedly until a determinate form emerges, the limit will be computed.

Solve Problem b

Taking the natural logarithm of L gives \(\ln(L) = x \ln \left(1-\frac{4}{x}\right)\) which is in the form `0 * infinity`. We can convert this expression into a `0/0` form by writing it as \(\frac{\ln \left(1-\frac{4}{x}\right)}{1/x}\). Now we can apply L'Hopital's rule until we get a determinate form to find the limit of L.

Solve Problem c

In order to determine the convergence of the series, the root test is used here as it is more effective given the form of the series. The root test involves taking the nth root of the absolute value of the general term and taking the limit as n goes to infinity. If this limit is less than 1, the series converges; if it is greater than 1, it diverges; otherwise, the test is inconclusive.

Solve Problem d

We recognize that the series given is a telescoping series. The Nth partial sum of the series can be written by adding up the first N terms. The terms will cancel out in such a way that only a few terms remain after cancellation. To find the sum of the series, compute the limit of the Nth partial sum as N goes to infinity.

Key concepts

L'Hopital's Rule

L'Hopital's Rule is an essential technique in calculus used to evaluate limits that initially appear to be in indeterminate forms like \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\). When approaching such a limit, the rule allows you to differentiate the numerator and denominator independently until a determinate form is reached. Once a determinate form is obtained, the limit can be evaluated directly using simple arithmetic.

This technique simplifies resolving complex limit problems by resorting to differentiation rules rather than algebraic manipulation. For example, in the cases from the exercise, L'Hopital's Rule was applied to transform complex logarithmic and exponential expressions into forms that could be easily evaluated. It's critical to ensure that the functions involved are continuous and differentiable within the domain of interest when applying L'Hopital's Rule.

Limit Evaluation

In calculus, evaluating limits is a fundamental process where we determine the value that a function approaches as the input approaches a particular point. Limits are essential in defining and understanding derivatives and integrals.

To evaluate limits, one can use various techniques, including factorization, rationalization, and the use of trigonometric limits. In the exercise, we faced limits as both \(\infty\) multiplied by zero and unconventional fractions. Converting these forms into something manageable, such as moving to a \(\frac{0}{0}\) form and using L'Hopital's Rule, helps in obtaining the actual limit.

• First, recognize the indeterminate form.
• Convert to an applicable standard form.
• Apply relevant calculus techniques or rules.
• Solve the resulting expression.

Proper limit evaluation is crucial as it ties directly into determining the behavior of functions at points of interest, discussing continuity, and ultimately drawing conclusions about the function's trend.

Series Convergence

Series convergence involves determining whether an infinite series reaches a finite value as the number of terms increases indefinitely. A series is a sum of terms of a sequence, and it converges if adding more terms brings the total closer to a specific number. If it doesn't stabilize to a finite number, the series diverges.

In calculus, tests like the root test or ratio test help determine convergence. Applying these tests involves checking whether the series terms decrease in size sufficiently fast as the series progresses.

• The root test, for instance, requires taking the nth root of the terms and evaluating their limit.
• If the outcome is less than one, the series converges. Otherwise, if greater than one, it diverges.

In our task, the root test provided insights into the series' behavior, helping determine the convergence by evaluating the limiting behavior of its terms.

Telescoping Series

A telescoping series is an infinite sum where subsequent terms cancel out previous terms, leading to a simpler expression. The structure of a telescoping series typically allows for quick summation because you end up with only a few terms remaining after cancellation.

To analyze a telescoping series, you identify patterns within the sequence that lead to cancellation. Often, this involves recognizing terms in the form of differences, such as \(a_n - a_{n+1}\). These cancel each other sequentially when summed together.

• The partial sums show this cancellation effect clearly.
• As you take the limit of the partial sum as it approaches infinity, you effortlessly find the sum of the series.

Telescoping is an efficient way to manage infinite series, turning a possibly daunting computational task into a straightforward arithmetic process.