Exercises

1. Give examples — other than those discussed in the text — of each of the following.
   1. A series that does not converge because its terms do not approach zero.
   2. A series that does not converge even though its terms do approach zero.
   3. A series that does converge.
2. For each of the following series, decide whether the series converges or diverges, and state how you know.

   a.   `sum_(k=1)^oo (text[(]-1text[)]^k)/(1+k)`	b.   `sum_(k=1)^oo (text[(]-1text[)]^k)/k`	c.   `sum_(k=1)^oo 1/(k+3)`
   d.   `sum_(k=1)^oo (text[(]-1text[)]^k k)/(k+1)`	e.   `sum_(k=1)^oo sin k`	f.   `sum_(k=1)^oo (text[(]-1text[)]^k)/(2k+1)`

3. For each of the following series, give its value if there is one and you know it. If the series converges, but you don't know its value, estimate the value with your computer or calculator.

   a.   `sum_(k=1)^oo (text[(]-1text[)]^k)/(1+k)`	b.   `sum_(k=1)^oo (text[(]-1text[)]^k)/k`	c.   `sum_(k=1)^oo 1/(k+3)`
   d.   `sum_(k=1)^oo (text[(]-1text[)]^k k)/(k+1)`	e.   `sum_(k=1)^oo sin k`	f.   `sum_(k=1)^oo (text[(]-1text[)]^k)/(2k+1)`

4. Ask a music major or musician in your class (yourself if you are a music major or musician) what the fractions `1`/`k` have to do with harmonics.
5. 1. Substitute `x=-1` into the arctangent series. How is the resulting series related to the Leibniz series?
   2. Does the series in part (a) converge or diverge? How do you know? If it converges, what is its value?
   3. Is the arctangent function even, odd, or neither? Are its Taylor polynomials even, odd, or neither?
   4. What does part (c) have to do with parts (a) and (b)?
6. What is the sum of the series `1+1/sqrt(2)+1/sqrt(3)+1/sqrt(4)+cdots`? Explain carefully. (Hint: Compare the terms and partial sums with those of the harmonic series.)
7. Does the series `1/(1text[,]000text[,]000)+1/(1text[,]000text[,]001)+1/(1text[,]000text[,]002)+cdots` converge or diverge? Explain carefully.
8. Does the series `1/(1text[,]000text[,]000)-1/(1text[,]000text[,]001)+1/(1text[,]000text[,]002)-cdots` converge or diverge? Explain carefully.