Advanced Calculus Single Variable

5.5 Exercises

1. Determine whether the following series converge absolutely, conditionally, or not at all and give reasons for your answers.
   1. ∑ _n=1^∞
      (− 1)
      ⁿ
      2n+n -n2n-
   2. ∑ _n=1^∞
      (− 1)
      ⁿ
      2nn2+2nn
   3. ∑ _n=1^∞
      (− 1)
      ⁿ
      -n-- 2n+1
   4. ∑ _n=1^∞
      (− 1)
      ⁿ
      10n- n!
   5. ∑ _n=1^∞
      (− 1)
      ⁿ
      100 n1.01n
   6. ∑ _n=1^∞
      (− 1)
      ⁿ
      3n3 n
   7. ∑ _n=1^∞
      (− 1)
      ⁿ
      n3 3n
   8. ∑ _n=1^∞
      (− 1)
      ⁿ
      3 nn!
   9. ∑ _n=1^∞
      (− 1)
      ⁿ
      -n! n100
2. Suppose ∑ _n=1^∞a_n converges. Can the same thing be said about ∑ _n=1^∞a_n²? Explain.
3. A person says a series converges conditionally by the ratio test. Explain why his statement is total nonsense.
4. A person says a series diverges by the alternating series test. Explain why his statement is total nonsense.
5. Find a series which diverges using one test but converges using another if possible. If this is not possible, tell why.
6. If ∑ _n=1^∞a_n and ∑ _n=1^∞b_n both converge, can you conclude the sum, ∑ _n=1^∞a_nb_n converges?
7. If ∑ _n=1^∞a_n converges absolutely, and b_n is bounded, can you conclude ∑ _n=1^∞a_nb_n converges? What if it is only the case that ∑ _n=1^∞a_n converges?
8. Prove Theorem 5.3.4. Hint: For ∑ _n=1^∞
   (− 1)
   ⁿb_n, show the odd partial sums are all no larger than ∑ _n=1^∞
   (− 1)
   ⁿb_n and are increasing while the even partial sums are at least as large as ∑ _n=1^∞
   (− 1)
   ⁿb_n and are decreasing. Use this to give another proof of the alternating series test. If you have trouble, see most standard calculus books.
9. Use Theorem 5.3.4 in the following alternating series to tell how large n must be so that
   | | ||∑ ∞k=1(− 1)kak − ∑nk=1 (− 1)kak||
   is no larger than the given number.
   1. ∑ _k=1^∞
      (− 1)
      ^k
      1k
      ,.001
   2. ∑ _k=1^∞
      (− 1)
      ^k
      12 k
      ,.001
   3. ∑ _k=1^∞
      (− 1)
      ^k−1
      1 √k-
      ,.001
10. Consider the series ∑ _k=0^∞
    (− 1)
    ⁿ
    --1-- √n+1
    . Show this series converges and so it makes sense to write
    (∑ ) ∞k=0 (− 1)n√n1+1-
    ². What about the Cauchy product of this series? Does it even converge? What does this mean about using algebra on infinite sums as though they were finite sums?
11. Verify Theorem 5.4.8 on the two series ∑ _k=0^∞2^−k and ∑ _k=0^∞3^−k.
12. You can define infinite series of complex numbers in exactly the same way as infinite series of real numbers. That is w = ∑ _k=1^∞z_k means: For every ε > 0 there exists N such that if n ≥ N, then
    ∑ |w − nk=1 zk|
    < ε. Here the absolute value is the one which applies to complex numbers. That is,
    |a + ib|
    =
    √ ------ a2 + b2
    . Show that if
    {an}
    is a decreasing sequence of nonnegative numbers with the property that lim_n→∞a_n = 0 and if ω is any complex number which is not equal to 1 but which satisfies
    |ω|
    = 1, then ∑ _n=1^∞ωⁿa_n must converge. Note a sequence of complex numbers,
    {an + ibn}
    converges to a + ib if and only if a_n → a and b_n → b. See Problem 6 on Page 138. There are quite a few things in this problem you should think about.
13. Suppose lim_k→∞s_k = s. Show it follows lim_n→∞
    1n
    ∑ _k=1ⁿs_k = s.
14. Using Problem 13 show that if ∑ _j=1^∞
    aj j-
    converges, then it follows

    ∑n lim 1- aj = 0. n→ ∞ nj=1

15. Show that if
    {pi}
    _i=1^∞ are the prime numbers, then ∑ _i=1^∞
    1- pi
    = ∞. That is, there are enough primes that the sum of their reciprocals diverges. Hint: Let ϕ
    (n)
    denote the number of primes less than equal to n. Then explain why

    n∑ 1 ϕ∏(n)( 1 ) ϕ∏(n) ∑ ϕ(n) --≤ 1 + -- ≤ e1∕pk = e k=1 1∕pk k=1 k k=1 pk k=1

    and consequently why lim_n→∞ϕ

    (n)
    = ∞ and ∑ _i=1^∞
    1- pi
    = ∞. You supply the details for the above.