Calculus I, II, and III: A Problem-Based Approach with Early Transcendentals:

Problem 6.30.

Write down the simplest expression you can find for the \(N^{th}\) partial sum, \(S_N,\) for the series \(1+1+1+ \dots\) and compute \(\dsp \lim_{N \to \infty} S_N.\)

Problem 6.31.

Write down the simplest expression you can find for the \(N^{th}\) partial sum, \(S_N,\) for the series \(\dsp 1 + 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \dots\) and compute \(\dsp \lim_{N \to \infty} S_N.\)

Problem 6.34.

Let \(\dsp a_0 = 1-\frac{1}{2}\text{,}\) \(\dsp a_1 = \frac{1}{2} - \frac{1}{3}\text{,}\) \(\dsp a_2 = \frac{1}{3} - \frac{1}{4}\text{,}\) \(\cdots\text{.}\)

1. Find a formula for the \(n^{th}\) term of the sequence, \(a_n.\)

2. Compute the limit of the terms of the sequence, \(a_n.\)

3. Find a formula for the \(N^{th}\) partial sum of the sequence, \(S_N.\)

4. Compute the limit of the partial sums, \(\displaystyle{\lim_{N \to \infty} S_N = \sum_{n=1}^{\infty} a_n}\text{.}\)

Problem 6.35.

Let \(\dsp a_0 = 3\text{,}\) \(\dsp a_1 = 3 (\frac{1}{2})\text{,}\) \(\dsp a_2 = 3(\frac{1}{2})^2\text{,}\) \(\cdots\text{.}\)

1. Find a formula for the \(n^{th}\) term, \(a_n\text{.}\)

2. Compute the limit of the terms, \(\displaystyle{\lim_{n \to \infty} a_n}\text{.}\)

3. Find a formula for the \(N^{th}\) partial sum, \(S_N\text{.}\)

4. Compute the limit of the partial sums, \(\displaystyle{\lim_{N \to \infty} S_N}\) or \(\displaystyle{\sum_{n=0}^{\infty} a_n}\text{.}\)

Problem 6.36.

Let \(\dsp a_0 = 5\text{,}\) \(a_1 = \frac{5}{3}\text{,}\) \(a_2 = \frac{5}{9}\text{,}\) \(\cdots\text{.}\)

1. Find a formula for the \(n^{th}\) term, \(a_n\text{.}\)

2. Compute the limit of the terms, \(\displaystyle{\lim_{n \to \infty} a_n}\text{.}\)

3. Find a formula for the \(N^{th}\) partial sum, \(S_N\text{.}\)

4. Compute the limit of the partial sums, \(\displaystyle{\lim_{N \to \infty} S_N}\) or \(\displaystyle{\sum_{n=0}^{\infty} a_n}\text{.}\)

Problem 6.37.

Let \(\dsp a_0 = a\text{,}\) \(a_1 = ar\text{,}\) \(a_2 = ar^2\text{,}\) \(\cdots\text{.}\)

1. Find a formula for the \(n^{th}\) term, \(a_n\text{.}\)

2. Compute the limit of the terms, \(\displaystyle{\lim_{n \to \infty} a_n}\text{.}\)

3. Find a formula for the \(N^{th}\) partial sum, \(S_N\text{.}\)

4. Compute the limit of the partial sums, \(\displaystyle{\lim_{N \to \infty} S_N}\) or \(\displaystyle{\sum_{n=0}^{\infty} a_n}\) where \(r\lt 1.\) (What happens when \(r = 1\) or \(r > 1\text{?}\)).

Problem 6.38.

Determine whether these geometric series converge and if so to what number.

1. \(\dsp{ \sum_{n=3}^\infty \frac{3}{4^n} }\)

2. \(\dsp{ \sum_{n=3}^\infty \frac{2^n}{3^{n+4}} }\)

3. \(\dsp{ \sum_{n=2}^\infty \frac{4^n}{3^n} }\)

4. \(\dsp{ \sum_{n=3}^\infty \frac{\pi}{e^{n-1}} }\)

Problem 6.39.

If a banker tells you that you will earn 6% interest compounded monthly, then you will actually earn \(\frac{6}{12}\)% each month, which is better than 6% annually since you get interest on your interest. Suppose you save $300 per month and earn \(\frac{6}{12}\)% interest on the amount of money in the bank at the end of each month. How much money do you have in 30 years? How much was savings? How much was interest?

Problem 6.41.

Create a decreasing sequence of positive terms \(a_1, a_2, a_3, a_4, \dots\) so that \(\dsp{ \lim_{n \to \infty} a_n}\) is some positive number, \(L\text{.}\) Explain why this series \(\displaystyle{\sum_{n=k}^{\infty} a_n}\) must diverge.

Problem 6.42.

In your mechanical engineering lab, you create a ball with a coefficient of elasticity so that if you dropped the ball from a height of one foot then the ball would bounce exactly to height \(\dsp \frac{1}{2n}\) feet on the \(n^{th}\) bounce. How far does the ball travel?

Problem 6.43.

The Harmonic Series.

Let \(\dsp a_1 = 1, a_2 = \frac{1}{2}, a_3 = \frac{1}{3}, \dots\text{.}\)

1. Find \(\displaystyle{\lim_{n \to \infty} a_n}\text{.}\)

2. Let \(\dsp f(x) = \frac{1}{x}\) be defined on \([1, \infty)\) and sketch a graph of both \(f\) and our sequence, \(a_n = \frac{1}{n}\) for \(n=1,2,3,\dots\) on the same coordinate axes.

3. Sketch rectangles (Riemann sums) whose areas approximate the area under the curve, \(f\text{.}\)

4. Use your picture to show that \(\displaystyle{S_N = \sum_{n=1}^{N-1} a_n \ge \int_1^N \frac{1}{x} \; \; dx}\text{.}\)

5. Show that \(\dsp \int_1^\infty \frac{1}{x} \; \; dx\) diverges, so \(\dsp \sum_{n=1}^{\infty} \frac{1}{n}\) diverges.

Problem 6.44.

Let \(\dsp a_n = \frac{1}{n^2}\) for \(n=1,2,3,\dots\text{.}\)

1. Find \(\displaystyle{\lim_{n \to \infty} a_n}\text{.}\)

2. Use a graph to show that for each \(N,\)

   \begin{equation*} \displaystyle{S_N = \sum_{n=1}^N \frac{1}{n^2} \le 1 + \int_1^N \frac{1}{x^2} \; \; dx \le 1 + \int_1^{\infty} \frac{1}{x^2} \; \; dx} \end{equation*}

   so that the sequence of partial sums \(\{S_n \}_{n=1}^{\infty}\) is bounded above.

3. Conclude that \(\displaystyle{\sum_{n=1}^{\infty} a_n}\) converges because \(\{S_n \}_{n=1}^{\infty}\) satisfies the hypothesis of Theorem 6.24.

Problem 6.46.

Apply the integral test to each series to determine whether it converges or diverges.

1. \(\dsp{ \sum_{n=4}^\infty \frac{1}{4n^3} }\)

2. \(\dsp{\sum_{n=2}^\infty \frac{3}{n^2} }\)

3. \(\dsp{ \sum_{n=1}^\infty \frac{5}{n^{1.001}} }\)

4. \(\dsp{ \sum_{n=1}^\infty \frac{4}{\sqrt{2n}} }\)

Problem 6.47.

Prove the integral test as follows. Let \(f\) be a continuous function such that \(f(x) \ge 0\) and is decreasing for all \(x \ge 1\text{,}\) and let \(f(n)=a_n\text{.}\) Suppose \(\dsp{ \int_1^\infty f(x) \ dx}\) converges.

1. Partition the interval \([1, N]\) into \(N-1\) equal intervals and sketch the rectangles as you did in Problem 6.43.

2. Express the total area of the rectangles in sigma notation; express the total area of the circumscribed rectangles in sigma notation.

3. Establish these two inequalities geometrically.

   \begin{equation*} \displaystyle{\sum_{n=1}^N a_n \le a_1 + \int_1^N f(x) \; dx} \; \mbox{ and } \displaystyle{\int_1^N f(x) \; \; dx \le \sum_{n=1}^N a_n} \end{equation*}

4. Prove that \(\displaystyle{\sum_{n=1}^{\infty} a_n}\) converges by showing that \(S_n\) is bounded above and increasing.

Problem 6.49.

Use the integral test to prove the p-series test for the case when \(p>1\text{.}\)

Problem 6.50.

Determine whether each series converges. If it does, use Problem 6.47 to give an upper bound for the infinite sum.

1. \(\dsp{ \sum_{n=4}^\infty \frac{5}{n^4} }\)

2. \(\dsp{\sum_{n=1}^\infty \frac{1}{2n^3} + \frac{2}{n^2} }\)

3. \(\dsp{\sum_{n=1}^\infty \frac{1}{\sqrt{2n^3}} }\)

Problem 6.51.

Consider the series \(\displaystyle{\sum_{n=1}^{\infty} \frac{1}{n^3+2}}.\)

1. Show that for all \(n \ge 1,\) we have \(\dsp \frac{1}{n^3+2} > 0.\)

2. Show that for all \(n \ge 1,\) we have \(\dsp \frac{1}{n^3+2} \lt \frac{1}{n^3}.\)

3. Why is this series monotonic and bounded and therefore convergent?

Problem 6.53.

Apply the direct comparison test to each of these series.

1. \(\dsp{ \sum_{n=2}^\infty \frac{1}{3^n+5} }\)

2. \(\dsp{ \sum_{n=2}^\infty \frac{1}{3^n-5} }\)

3. \(\dsp{\sum_{n=3}^\infty \frac{3}{n^2 \ln(n)} }\)

4. \(\dsp{\sum_{n=3}^\infty \frac{e}{5(n^5 + 32)} }\)

Problem 6.56.

Apply the limit comparison test to each series to determine if it converges or diverges.

1. \(\dsp{ \sum_{n=2}^\infty \frac{1}{3^n-5} }\)

2. \(\dsp{ \sum_{n=3}^\infty \frac{3}{n + \pi} }\)

3. \(\displaystyle{\sum_{n=1}^{\infty} \sin \Big(\frac{1}{n}\Big)}\) What function does $\sin(x)$ look like near $0$?

4. \(\displaystyle{\sum_{n=1}^{\infty} \Big(\frac{n+1}{2n+3}\Big) \Big(\frac{3^n}{5^n + 1}\Big)}\)

Problem 6.57.

In the real world of computer science and operations research, the “average” of a series is an important notion. Suppose we have a sequence, \(\{ a_n \}_{n=1}^\infty\) where \(0 \leq a_n \leq 1\) for all positive integers, \(n.\)

Consider the sequence given by

1. Suppose \(a_1=a_2=a_3=\dots = 1\) Does the sequence \(\{ A_N \}_{N=1}^\infty\) converge? To what?

2. Suppose \(a_1=1, a_2 =0, a_3= 1, \dots\) Does the sequence \(\{ A_N \}_{N=1}^\infty\) converge? To what?

3. Do you think that \(\{ A_N \}_{N=1}^\infty\) converges no matter what \(a_1, a_2, \dots\) is? If so give an argument, if not, give an example for \(a_1, a_2, \dots\) and show that for this sequence the \(A_n\)'s do not converge.

Problem 6.58.

Determine the convergence or divergence of the following series by using any of the previously stated tests.

1. \(\displaystyle{\sum_{n=1}^{\infty} \frac{2}{3^n + n}}\)

2. \(\displaystyle{\sum_{n=1}^{\infty} \frac{n^2-n +1}{2n^2 -n - 3}}\)

3. \(\displaystyle{\sum_{n=4}^{\infty} \frac{-2}{n^2-4n+3}}\)

4. \(\displaystyle{\sum_{n=4}^{\infty} \frac{1}{n!}}\)

5. \(\displaystyle{\sum_{n=1}^{\infty} \cos(n \pi)}\)

Problem 6.59.

Determine the convergence or divergence of each of the following series. If any of them converge, compute the number to which it coverges.

1. \(\displaystyle{\sum_{n=1}^{\infty}{ {{|\sin(\frac{n \pi}{2})}|} \over {2^n}}}\)

2. \(\displaystyle{\sum_{n=1}^{\infty} (2n+1)^\frac{1}{n}}\)

3. \(\displaystyle{\sum_{n=2}^{\infty} \frac{n^2+2n+1}{3n-4}}\)

4. \(\displaystyle{\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{e^n}}\)