10.5.2 The Harmonic Series

The question of convergence of the harmonic series brings us face to face with subtlety at a level we have not experienced previously in this course. We are adding up terms that get smaller and smaller. After a million such terms, every new term must be smaller than one-millionth. After a billion terms, every term is smaller than one-billionth. Moreover, the partial sum of the first billion terms is only about `20`. (Don't try to check that with your computer or calculator — trust us.) How could such a sum not converge? But the fact is, it does not! In the next activity you can convince yourself that the sum of all the terms is larger than every positive integer; i.e., the sum is infinite!

Activity 2

1. Show that the harmonic series may be written in the form

   ∑ k = 1 ∞ 1 k .

2. Show that

   1 3 + 1 4 > 1 2 ,

   and use this to conclude that

   1 + 1 2 + 1 3 + 1 4 > 4 2 .

   (Hint: You don't need a computer or calculator for this.)

3. Show that

   1 5 + 1 6 + 1 7 + 1 8 > 1 2 ,

   and use this to show that

   ∑ k = 1 8 1 k > 5 2 .

4. Find an integer `n` so that

   ∑ k = 1 n 1 k > 6 2 .

5. Explain why, for any positive integer `r`, there is an integer `n` such that

   ∑ k = 1 n 1 k > r 2 .

6. Explain why the harmonic series does not converge.

7. Does this contradict the Divergence Test? Why or why not?

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