Chapter 11

For which of the following series is the Ratio Test inconclusive (that is, if fails to give a definite answer)? (a) \( \displaystyle \sum_{n = 1}^{\infty} \frac {1}{n^3} \) (b) \( \displaystyle \sum_{n = 1}^{\infty} \frac {n}{n^2} \) (c) \( \displaystyle \sum_{n = 1}^{\infty} \frac {( - 3)^{n-1}}{\sqrt{n}} \) (d) \( \displaystyle \sum_{n = 1}^{\infty} \frac {\sqrt{n}}{1 + n^2} \)

Show that if \( a_n > 0 \) and \( \lim_{n \to \infty} na_n \not= 0, \) then \( \sum a_n \) is divergent.

Determine whether the series is convergent or divergent by expressing \( s_n \) as a telescoping sum (as in Examples 8). If it is convergent, find its sum. \( \displaystyle \sum_{n = 2}^{\infty} \frac {2}{n^2 - 1} \)

Determine whether the sequence converges or diverges. If it converges, find the limit. \( a_n = \frac { \cos^2 n}{2^n} \)

For which positive integers \( k \) is the following series convergent? \( \displaystyle \sum_{n = 1}^{\infty} \frac {(n!)^2}{(kn)!} \)

Show that if \( a_n > 0 \) and \( \sum a_n \) is convergent, then \( \sum \ln(1 + a_n) \) is convergent.

Determine whether the sequence converges or diverges. If it converges, find the limit. \( a_n = \sqrt [n]{2^{1 + 3n}} \)

Determine whether the series is convergent or divergent by expressing \( s_n \) as a telescoping sum (as in Examples 8). If it is convergent, find its sum. \( \displaystyle \sum_{n = 1}^{\infty} \ln \frac {n}{n + 1} \)

Find the Maclaurin series of \( f \) (by any method) and its radius of convergence. Graph \( f \) and its first few Taylor polynomials on the same screen. What do you notice about the relationship between these polynomials and \( f? \) \( f(x) = \cos (x^2) \)

(a) Show that \( \sum_{n=0}^{\infty} x^n/n! \) converges for all \( x. \) (b) Deduce that \( lim_{n \to \infty} x^n/n! = 0 \) for all \( x. \)

If \( \sum a_n \) is a convergent series with positive terms, is it true that \( \sum \sin(a_n) \) is also convergent?

Find all positive values of \( b \) for which the series \( \sum_{n = 1}^{\infty} b^{\ln n} \) converges.

Determine whether the series is convergent or divergent by expressing \( s_n \) as a telescoping sum (as in Examples 8). If it is convergent, find its sum. \( \displaystyle \sum_{n = 1}^{\infty} \frac {3}{n(n + 3)} \)

Determine whether the sequence converges or diverges. If it converges, find the limit. \( a_n = n \sin (1/n) \)

Find the Maclaurin series of \( f \) (by any method) and its radius of convergence. Graph \( f \) and its first few Taylor polynomials on the same screen. What do you notice about the relationship between these polynomials and \( f? \) \( f(x) = \ln (1 + x^2) \)

Let \( \sum a_n \) be a series with positive terms and let \( r_n = a_{n+1} / a_n. \) Suppose that \( lim_{n \to \infty} r_n = L < 1, \) so \( \sum a_n \) converges by the Ratio Test. As usual, we let \( R_n \) be the remainder after \( n \) terms, that is, \( R_n = a_{n+1} + a_{n+2} + a_{n+3} + \cdot \cdot \cdot \) (a) If \( \\{r_n\\} \) is a decreasing sequence and \( r_{n+1} < 1, \) show, by summing a geometric series, that \( R_n \le \frac {a_{n+1}}{1 - r_{n+1}} \) (b) If \( \\{r_n\\} \) is a decreasing sequence, show that \( R_n \le \frac {a_{n+1}}{1 - L} \)

If \( \sum a_n \) and \( \sum b_n \) are both convergent series with positive terms, is it true that \( \sum a_n b_n \) is also convergent?

Find all values of \( c \) for which the following series converges. \( \displaystyle \sum_{n = 1}^{\infty} \left( \frac {c}{n} - \frac {1}{n + 1} \right) \)

Determine whether the sequence converges or diverges. If it converges, find the limit. \( a_n = 2^{-n} \cos n \pi \)

Find the Maclaurin series of \( f \) (by any method) and its radius of convergence. Graph \( f \) and its first few Taylor polynomials on the same screen. What do you notice about the relationship between these polynomials and \( f? \) \( f(x) = xe^{-x} \)

Determine whether the series is convergent or divergent by expressing \( s_n \) as a telescoping sum (as in Examples 8). If it is convergent, find its sum. \( \displaystyle \sum_{n = 1}^{\infty} \left( e^{1/n} - e^{1/(n + 1)} \right) \)

Determine whether the sequence converges or diverges. If it converges, find the limit. \( a_n = \left( 1+ \frac {2}{n} \right)^n \)

Find the Maclaurin series of \( f \) (by any method) and its radius of convergence. Graph \( f \) and its first few Taylor polynomials on the same screen. What do you notice about the relationship between these polynomials and \( f? \) \( f(x) = \tan^{-1} (x^3) \)

Determine whether the sequence converges or diverges. If it converges, find the limit. \( a_n = \sqrt[n]{n} \)

Determine whether the series is convergent or divergent by expressing \( s_n \) as a telescoping sum (as in Examples 8). If it is convergent, find its sum. \( \displaystyle \sum_{n = 2}^{\infty} \frac {1}{n^3 - n} \)

Use the Maclaurin series for \( \cos x \) to compute \( \cos 5^o \) correct to five decimal places.

Prove the Root Test. [Hint for part (i): Take any number \( r \) such that \( L < r < 1 \) and use the fact that there is an integer \( N \) such \( \sqrt [n]{\mid a_n \mid} < r \) whenever \( n \ge N.] \)

Let \( x = 0.99999 . . . . \) (a) Do you think that \( x < 1 \) or \( x = 1? \) (b) Sum a geometric series to find the value of \( x. \) (c) How many decimal representations does the number 1 have? (d) Which numbers have more than one decimal representation?

Determine whether the sequence converges or diverges. If it converges, find the limit. \( a_n = \ln(2n^2 + 1) - \ln(n^2 + 1) \)

Use the Maclaurin series for \( e^x \) to calculate \( 1/\sqrt[10]{e} \) correct to five decimal places.

Around 1910, the Indian mathematician Srinivasa Ramanujan discovered the formula \( \frac {1}{\pi} = \frac {2 \sqrt{2}}{9801} \displaystyle \sum_{n = 0}^{\infty} \frac {(4n)!(1103 + 26390n)}{(n!)^4396^{4n}} \) William Gosper used this series in 1985 to compute the first 17 million digits of \( \pi . \) (a) Verify that the series is convergent. (b) How many correct decimal places of \( \pi \) do you get if you use just the first term of the series? What if you use two terms?

Determine whether the sequence converges or diverges. If it converges, find the limit. \( a_n = \frac { (\ln n)^2}{n} \)

(a) Use the binomial series to expand \( 1/\sqrt {1 - x^2}. \) (b) Use part (a) to find the Maclaurin series for \( \sin^{-1} x. \)

Given any series \( \sum a_n, \) we define a series \( \sum a_{n}^{+} \) whose terms are all the positive terms of \( \sum a_n, \) and a series \( \sum a_{n}^{-} \) whose terms are all the negative terms of \( \sum a_{n.} \) To be specific, we let \( a_{n}^{+} = \frac {a_n + \mid a_n \mid }{2} \) \( a_{n}^{-} = \frac {a_n - \mid a_n \mid }{2} \) Notice that if \( a_n > 0 \), then \( a_{n}^{+} = a_n \) and \( a_{n}^{-} = 0 \) whereas if \( a_n < 0, \) then \( a_{n}^{-} = a_n \) and \( a_{n}^{+} = 0. \) (a) If \( \sum a_n \) is absolutely convergent, show that both of the series \( \sum a_{n}^{+} \) and \( \sum a_{n}^{-} \) are convergent. (b) If \( \sum a_n \) is conditionally convergent, show that both of the series \( \sum a_{n}^{+} \) and \( \sum a_{n}^{-} \) are convergent.

Express the number as a ratio of intergers. \( 0. \overline 8 = 0.8888 . . . \)

Determine whether the sequence converges or diverges. If it converges, find the limit. \( a_n = \arctan (\ln n) \)

(a) Expand \( 1/\sqrt[4]{1 + x} \) as a power series. (b) Use part (a) to estimate \( 1/\sqrt [4]{1.1} \) correct to three decimal places.

Express the number as a ratio of integers. \(0 . \overline{46}=0.46464646 \ldots\)

Determine whether the sequence converges or diverges. If it converges, find the limit. \( a_n = n - \sqrt {n + 1} \sqrt {n + 3} \)

Evaluate the indefinite integral as an infinite series. \( \int \sqrt {1 + x^3} dx \)

Express the number as a ratio of intergers. \( 2. \overline {516} = 2.516516516 . . . \)

Determine whether the sequence converges or diverges. If it converges, find the limit. \( \left \\{ 0, 1, 0, 0, 1, 0, 0, 0, 1, . . . \right \\} \)

Suppose the series \( \sum a_n \) is conditionally convergent. (a) Prove that the series \( \sum n^2 a_n \) is divergent. (b) Conditional convergence of \( \sum a_n \) is not enough to determine whether \( \sum na_n \) is convergent. show this by giving an example of conditionally convergent series such that \( \sum na_n \) converges and an example where \( \sum na_n \) diverges.

Evaluate the indefinite integral as an infinite series. $$ \int x^2 \sin (x^2) dx $$

Express the number as a ratio of intergers. \( 10.1 \overline {35} = 10.135353535 . . . \)

Determine whether the sequence converges or diverges. If it converges, find the limit. \( \left \\{ \frac {1}{1}, \frac {1}{3}, \frac {1}{2}, \frac {1}{4}, \frac {1}{3}, \frac {1}{5}, \frac {1}{4}, \frac {1}{6}, . . . \right \\} \)

Express the number as a ratio of intergers. \( 1.234 \overline {567} \)

Determine whether the sequence converges or diverges. If it converges, find the limit. \( a_n = \frac {n!}{2^n} \)

Evaluate the indefinite integral as an infinite series. \( \int \frac {\cos x - 1}{x} dx \)

Evaluate the indefinite integral as an infinite series. $$ \int \arctan (x^2) dx $$