Let an = Hn ln n, where Hn is the nth harmonic number: Hn = 1 + 1 2 + 1 3 ++ 1 n (a)
Chapter 10, Problem 89(choose chapter or problem)
Let an = Hn ln n, where Hn is the nth harmonic number: Hn = 1 + 1 2 + 1 3 ++ 1 n (a) Show that an 0 for n 1. Hint: Show that Hn n+1 1 dx x . (b) Show that {an} is decreasing by interpreting an an+1 as an area. (c) Prove that lim n an exists. This limit, denoted , is known as Eulers Constant. It appears in many areas of mathematics, including analysis and number theory, and has been calculated to more than 100 million decimal places, but it is still not known whether is an irrational number. The first 10 digits are 0.5772156649.
Unfortunately, we don't have that question answered yet. But you can get it answered in just 5 hours by Logging in or Becoming a subscriber.
Becoming a subscriber
Or look for another answer