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Chapter 10 Challenge Problems
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Chapter 10 Challenge Problems

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\(\hspace{0.56em}1.\)
Using a Proof by Contradiction, prove that the sequence defined by \[a_1 = 3 \cmaa a_{n + 1} = -\frac{1}{a_n} \cma\] is divergent.
\(\hspace{0.56em}2.\)
Consider the family of sums \[S = \sum_{n = 1}^\infty a_n + \sum_{n = 1}^\infty b_n \pd\]
  1. Find formulas for \(a_n\) and \(b_n\) such that \(\sum_{n = 1}^\infty a_n\) and \(\sum_{n = 1}^\infty b_n\) both diverge and \(S\) diverges.
  2. Find formulas for \(a_n\) and \(b_n\) such that \(\sum_{n = 1}^\infty a_n\) and \(\sum_{n = 1}^\infty b_n\) both diverge and \(S\) converges.
\(\hspace{0.56em}3.\)
Let \(a_n\) and \(b_n\) be positive functions such that \(\sum b_n\) diverges, and let \[L = \lim_{n \to \infty} \frac{a_n}{b_n} \pd\] Prove that the Limit Comparison Test is inconclusive if \(L = 0.\)
\(\hspace{0.56em}4.\)
When banks provide loans, a country's money supply increases. Banks loan money to other banks, but banks are required to hold a certain proportion—called the reserve ratio \((R)\)—of their total assets. Suppose a deposit of \(x\) dollars is made to a bank account. To maximize its revenue, the bank holds on to \(xR\) dollars and lends out \(x(1 - R)\) dollars to another bank. The new bank then lends out \(x(1 - R)^2\) dollars to a third bank, and this repetition leads to rapid multiplication of the initial deposit of \(x\) dollars.
  1. Assume the initial deposit of \(x\) dollars is already part of the money supply. Based on an infinite series, derive an expression for \(M,\) the increase in the money supply (the money generated) when an initial deposit of \(x\) dollars is made in a country whose reserve ratio is \(R.\)
  2. During times of inflation or rapid growth, the reserve ratio can be increased to slow the growth of the money supply, thus combating inflation. Determine, in terms of \(x\) and \(R,\) how much less money is added to the money supply when \(R\) is increased by \(0.1.\)
  3. The reserve ratio is often decreased during times of recession to encourage economic growth. In 2020, amid the onset of quarantine and lockdowns due to the COVID-19 pandemic, the Federal Reserve (the Fed) dropped the reserve ratio to \(0.\) Calculate and interpret \(\lim_{R \to 0^+} M.\)
\(\hspace{0.56em}5.\)
To simulate an object slowing down due to drag or friction forces, a video game developer may program the object's velocity to decrease by \(1\%\) every frame. (For example, the code may be speed = 0.99 * speed.) Suppose that a video game operates at \(60\) frames per second, and consider a particle whose initial speed is \(10\) meters per second. Assume the object's position is updated once per frame using its current velocity.
  1. After \(1\) second, what is the particle's new speed?
  2. Using an infinite geometric series, calculate the total distance \(D_f\) the particle travels.
  3. The developer programs any moving object to disappear once its speed becomes less than \(2\%\) of its initial speed. Under this condition, what distance \(D_e\) does the particle travel?
  4. What percentage of the distance \(D_f\) does \(D_e\) represent?
\(\hspace{0.56em}6.\)
An infinite geometric series converges to a positive number \(S.\) If the second term in the series is \(2,\) then what is the smallest possible value of \(S \ques\)
\(\hspace{0.56em}7.\)
Calculate the exact value of \[\tfrac{1}{4} + \tfrac{2}{3} + \tfrac{5}{6} + \tfrac{1}{16} - \tfrac{2}{9} - \tfrac{5}{12} + \tfrac{1}{64} + \tfrac{2}{27} + \tfrac{5}{24} + \cdots \pd\]
\(\hspace{0.56em}8.\)
The Harmonic series diverges very slowly. An untrained student may fallaciously conclude that it converges after examining its partial sums.
  1. Interpreting the Harmonic series as a right-endpoint approximation with \(\Delta x = 1\) for the area under \(y = 1/x\) from \(x = 1\) to \(x = N,\) show that \[\sum_{i = 1}^N \frac{1}{i} \leq 1 + \ln N \pd\]
  2. Using the result in part (a), show that the sum of the first million terms of the Harmonic series is less than \(15\) and that the sum of its first billion terms is less than \(22.\)
  3. Using the result in part (a), estimate how many terms in the Harmonic series are needed to attain a sum of \(50.\)
\(\hspace{0.56em}9.\)
Find the values of \(c\) such that \[\int_1^\infty \frac{1}{(x - c)^2} \di x \and \sum_{n = 1}^\infty \frac{1}{n^{c^2 - 3}}\] both converge.
\(10.\)
Show that \[\int_0^x \sum_{n = 1}^\infty \frac{(-1)^n \, t^{n - 1}}{(n - 1)!} \di t = e^{-x} - 1 \pd\]
\(11.\)
This exercise examines series of polynomials.
  1. Show that the sum of a nonzero polynomial is divergent.
  2. Show that the Ratio Test fails to prove that the sum of a polynomial is divergent.
  3. Let \(P(n)\) be a polynomial. Prove that \(\sum P(n) \, e^{-n}\) is absolutely convergent.
\(12.\)
For all \(x,\) \[f(x) = \sum_{n = 0}^\infty \frac{x^n}{n!} \pd\]
  1. Show that \(f\) satisfies the differential equation \(f'(x) = f(x).\)
  2. Show that \(f(x) = e^x.\)
  3. Determine a power series for \(e^{2x}.\)
\(13.\)
Let \(\{a_n\}\) and \(\{b_n\}\) be sequences such that \(b_n\) is decreasing to \(0\) and \(\abs{a_n}\) \(= b_n - b_{n + 1}\) for all \(n.\) Prove that \(\sum_{n = 1}^\infty a_n\) converges absolutely.
\(14.\)
Determine whether the following series converges or diverges: \[\sum_{n = 2}^\infty \int_1^\infty \frac{1}{x^n} \di x \pd\]