Calculus II, Worksheet 2 Name: Please answer the following ...

Calculus II, Worksheet 2Name:Please answer the following questions in the spaces provided, or on your own paper. You mayuse your textbook, but do not consult any other sources or with each other. This worksheetis due on July 30th. As you have plenty of time for this, you will not receive credit forillegible or excessively disorganized work.1. (20 pts) It is often difficult to find exact values for series, but with sufficient ingenuity,∞∑ (−1) k+1methods can be found. This exercise outlines one way to show that= ln 2.kn∑ 1(a) Define the sequence a n = − ln n. Use the Bounded Monotone Convergencekk=1Theorem to show that γ = lim a n exists. You don’t need to find its value, but itn→∞so happens that γ ≈∫ 0.57722.ndx(Hint: Note ln n =1 x . Interpret a n geometrically, similarly to the picture in#30 of Section 10.2)k=1n∑ (−1) k+1n∑ 1(b) Define s n =. Define h n =kk . Show that h n = h 2n − s 2n .k=1k=1(Hint: Start with the right-hand side and simplify to the left-hand side.)

(c) Note that algebraic manipulation of (b) givess 2n = (h 2n − ln(2n)) − (h n − ln n) + ln(2n) − ln n.Using the result from (a), show that(Hint: What is limn→∞s 2n ?)∞∑ (−1) k+1k=1k= ln 2.