Chapter 11

Determine whether the sequence is increasing, decreasing, or not monotonic. Is the sequence bounded? \( a_n = n(-1)^n \)

Find the sum of the series. \( \sum_{n = 0}^{\infty} \frac {3^n}{5^n n!} \)

Determine whether the sequence is increasing, decreasing, or not monotonic. Is the sequence bounded? \( a_n = 2 + \frac{(-1)^n}{n} \)

Find the value of c such that \( \displaystyle \sum_{n = 0}^{\infty} e^{nc} = 10 \)

Find the sum of the series. \( \sum_{n = 0}^{\infty} \frac {(-1)^n \pi^{2n + 1}}{4^{2n + 1}(2n + 1)!} \)

Determine whether the sequence is increasing, decreasing, or not monotonic. Is the sequence bounded? \( a_n = 3 - 2ne^{-n} \)

Find the sum of the series. \( 1 - \ln 2 + \frac {(\ln 2)^2}{2!} - \frac {(\ln 2)^3}{3!} + \cdot \cdot \cdot \)

Determine whether the sequence is increasing, decreasing, or not monotonic. Is the sequence bounded? \( a_n = n^3 - 3n + 3 \)

Find the sum of the series. \( 3 + \frac {9}{2!} + \frac {27}{3!} + \frac {81}{4!} + \cdot \cdot \cdot \)

Find the limit of the sequence \( \left\\{ \sqrt 2, \sqrt{2\sqrt2}, \sqrt{2\sqrt{2\sqrt2}}, \cdot \cdot \cdot \right\\} \)

A sequence \( \left\\{ a_n \right\\} \) is given by \( a_1 = \sqrt 2, a_{n + 1} = \sqrt {2 + a_n}. \) (a) By induction or otherwise, show that \( \left\\{ a_n \right\\} \) is increasing and bounded above by 3. Apply the Monotonic Sequence Theorem to show that \( \lim_{n\to\infty} a_n \) exists. (b) Find \( \lim_{n\to\infty} a_n. \)

What is wrong with the following calculation? \( 0 = 0 + 0 + 0 + \cdot \cdot \cdot \) \( = (1 - 1) + (1 - 1) + (1 - 1) + \cdot \cdot \cdot \) \( = 1 - 1 + 1 - 1 + 1 - 1 + \cdot \cdot \cdot \) \( = 1 + (-1 + 1) + (-1 + 1) + (-1 + 1) + \cdot \cdot \cdot \) \( = 1 + 0 + 0 + 0 + \cdot \cdot \cdot = 1 \) (Guido Ubaldus thought that this proved the existence of God because "something has been created out of nothing.")

Show that the sequence defined by \( a_1 = 1 \) \( a_{n + 1} = 3 - \frac{1}{a_n} \) is increasing and \( a_n < 3 \) for all \( n. \) Deduce that \( \\{ a_n \\} \) is convergent and find its limit.

If \(f(x)=\left(1+x^{3}\right)^{30},\) what is \(f^{(5)}(0) ?\)

Suppose that \( \sum_{n = 1}^{\infty} a_n \left( a_n \not= 0 \right) \) is known to be a convergent series. Prove that \( \sum_{n = 1}^{\infty} 1/a_n \) is a divergent series.

Show that the sequence defined by \( a_1 = 2 \) \( a_{n + 1} = \frac {1}{3 - a_n} \) satisfies \( 0 < a_n \le 2 \) and is decreasing. Deduce that the sequence is convergent and find its limit.

(a) Fibonacci posed the following : Suppose that rabbits live forever and that every month each pair produces a new pair which becomes productive at age 2 months. If we start with one newborn pair, how many pairs or rabbits will we have in the \( n \)th month? Show that the answer is \( f_n \) where \( \\{ f_n \\} \) is the Fibonacci sequence defined in Example 3(c). (b) Let \( a_n = f_{n + 1} / f_n \) and show that \( a_{n - 1} = 1 + 1/a_{n - 2}. \) Assuming that \( \\{ a_n \\} \) is convergent, find its limit.

Prove Taylor's Inequality for \( n = 2, \) that is, prove that if \( \mid f''' (x) \mid \le M \text { for } \mid x - a \mid \le d, \) then \( \mid R_2 (x) \mid \le \frac {M}{6} \mid x - a \mid^3 \) for \( \mid x - a \mid \le d \)

(a) Show that the function defined by \( f(x) = \left\\{\begin{array} (e^{-1/x^2} \text{ if } x \not= 0 \\\0 \text { if } x = 0 \end{array}\right. \) is not equal to its Maclaurin series. (b) Graph the function in part (a) and comment on its behavior near the origin.

If \( \sum a_n \) is divergent and \( c \not= 0, \) show that \( \sum ca_n \) is divergent.

(a) Let \( a_1 =a, a_2 = f(a), a_3 = f(a_2) = f( f(a)), . . . , a_{n + 1} = f(a_n), \) where \( f \) is a continuous function. If \( lim_{n \to\infty} a_n = L, \) show that \( f(L) = L. \) (b) Illustrate part (a) by taking \( f(x) = \cos x, a = 1, \) and estimating the value of \( L \) to five decimal places.

Use the following steps to prove (17). (a) Let \( g(x) = \sum_{n = 0}^{\infty} (^k_n) x^n. \) Differentiate this series to show that \( g'(x) = \frac {kg(x)}{1 + x} -1 < x < 1 \) (b) Let \( h(x) = (1 + x)^{-k} g(x) \) and show that \( h'(x) = 0. \) (c) Deduce that \( g(x) = (1 + x)^k. \)

If \( \sum a_n \) is convergent and \( \sum b_n \) is divergent, show that the series \( \sum \left( a_n + b_n \right) \) is divergent. [Hint: Argue by contradiction.]

If \( \sum a_n \) and \( \sum b_n \) are both divergent, is \( \sum \left(a_n + b_n \right) \) necessarily divergent?

Suppose that a series \( \sum a_n \) has positive terms and its partial sums \( s_n \) satisfy the inequality \( s_n \le 1000 \) for all \( n. \) Explain why \( \sum a_n \) must be convergent.

The Fibonacci sequence was defined in Section 11.1 by the equations \( f_1 = 1, f_2 = 1, f_n = f_{n -1} + f_{n - 2} n \ge 3 \) Show that each of the following statements is true. (a) \( \frac {1}{f_{n - 1} f_{n + 1}} = \frac {1}{f_{n - 1} f_n} - \frac {1}{f_n f_{n + 1}} \) (b) \( \displaystyle \sum_{n = 2}^{\infty} \frac {1}{f_{n - 1} f_{n + 1}} = 1 \) (c) \( \displaystyle \sum_{n = 2}^{\infty} \frac {f_n}{f_{n -1} f_{n + 1}} = 2 \)

The Cantor set, named after the German mathematician George Cantor (1845 - 1918), is constructed as follows. We start with the closed interval [0, 1] and remove the open interval \( \left( \frac {1}{3}, \frac {2}{3} \right). \) That leaves the two intervals \( \left[ 0, \frac {1}{3} \right] \) and \( \left[ \frac {2}{3}, 1 \right] \) and we remove the open middle third of each. Four intervals remain and given we remove the open middle third of each of them. We continue this procedure indefinitely, at each step removing the open middle third of every interval that remains from the preceding step. The Cantor set consists of the numbers that remain in [0, 1] after all those intervals have been removed. (a) Show that the total length of all the intervals that are removed is 1. Despite that, the Cantor set contains infinitely many numbers. Give examples of some numbers in the Cantor set. (b) The Sierpinski carpet is a two-dimensional counterpart of the Cantor set. It is constructed by removing the center one-ninth of a square of side 1, then removing the centers of the eight smaller remaining squares, and so on. (The figure shows the first three steps of the construction.) Show that the sum of the areas of the removed squares is 1. This implies that the Sierpinski carpet has area 0.

Prove that if \( \lim_{n \to \infty} a_n = 0 \) and \( \left \\{ b_n \right \\} \) is bounded, then \( \lim_{n \to\infty} (a_n b_n) = 0. \)

(a) A sequence \( \left\\{ a_n \right\\} \) is defined recursively by the equation \( a_n = \frac {1}{2} \left(a_{n - 1} + a_{n - 2} \right) \) for \( n \ge 3, \) where \( a_1 \) and \( a_2 \) can be any real numbers. Experiment with various values of \( a_1 \) and \( a_2 \) and use your calculator to guess the limit of the sequence. (b) Find \( \lim_{n \to \infty} a_n \) in terms of \( a_1 \) and \( a_2 \) by expressing \( a_{n + 1} - a_n \) in terms of \( a_2 - a_1 \) and summing a series.

Consider the series \( \sum_{n = 1}^{\infty} n/(n + 1)!. \) (a) Find the partial sums \( s_1, s_2, s_3, \) and \( s_4. \) Do you recognize the denominators? Use the pattern to guess a formula for \( s_n. \) (b) Use mathematical induction to prove your guess. (c) Show that the given infinite series is convergent, and find its sum.

Let \( a \) and \( b \) be positive numbers with \( a > b. \) Let \( a_1 \) be their arithmetic mean and \( b_1 \) their geometric mean: \( a_1 = \frac {a + b}{2} \) \( b_1 = \sqrt {ab} \) Repeat this process so that, in general, \( a^{n + 1} = \frac {a_n + b_n}{2} \) \( b_{n + 1} = \sqrt {a_n b_n} \) (a) Use mathematical induction to show that \( a_n > a_{n + 1} > b_{n + 1} > b_n \) (b) Deduce that both \( \\{ a_n \\} \) and \( \\{ b_n \\} \) are convergent. (c) Show that \( \lim_{n \to\infty} a_n = \lim_{n \to \infty} b_n \). Gauss called the common value of these limits the arithmetic-geometric mean of the numbers \( a \) and \( b. \)

(a) Show that if \( \lim_{n \to \infty} a_{2n} = L \) and \( \lim_{n \to\infty} a_{2n + 1} = L, \) then \( \\{ a_n \\} \) is convergent and \( \lim_{n \to \infty} a_n = L \). (b) If \( a_1 = 1 \) and \( a_{n + 1} = 1 + \frac {1}{1 + a_n} \) find the first eight terms of the sequence \( \\{ a_n \\} \). Then use part (a) to show that \( \lim_{n \to \infty} a_n = \sqrt{ 2 } \). This gives the continued fraction expansion \( \sqrt{ 2 } = 1 + \frac {1}{ 2 + \frac {1}{2 + \cdot \cdot \cdot}} \)

The size of an undisturbed fish population has been modeled by the formula \( p_{n + 1} = \frac {bp_n}{a + p_n} \) where \( p_n \) is the fish population after \( n \) years and \( a \) and \( b \) are positive constants that depend on the species and its environment. Suppose that the population in year 0 is \( p_0 > 0\. \) (a) Show that if \( \\{ p_n \\} \) is convergent, then the only possible values for its limit are 0 and \( b - a \). (b) Show that \( p_{n + 1} < (b/a)p_n \). (c) Use part (b) to show that if \( a > b, \) then \( \lim_{n \to \infty} p_n = 0 \); in other words, the population dies out. (d) Now assume that \( a < b \). Show that if \( p_0 < b - a \), then \( \\{ p_n \\} \) is increasing and \( 0 < p_n < b - a \). Show also that if \( p_0 > b - a \), then \( \\{ p_n \\} \) is decreasing and \( p_n > b - a \). Deduce that if \( a < b \), then \( \lim_{n \to \infty} p_n = b - a \).