Calculus Series
1. Show that if 0 S 51? S 1/2 then -2:17 S ln(1 – T) S -1?. Suggestion: Let
f(:17) : ln(1-:17). show that its first derivative is decreasing. calculate possible
derivative values for the X in question. invoke the l\~“lean Value Theorem.
2. Given a sequence an. Where each 0 S an S 1/2. form the infinite product
00
W1 – an)
71:1
Each term in the product is between 0 and 1. so the partial products are
monotone decreasing. Using the result of the first problem. show that
00 00
H<1 _ (1/71) 2 0 fi 5 (1/71 2 00
71:1 71:1
3. (Deep thinking for this one) Suppose that the positive series 2 an con-
verges. Define
bit 2 (Lg-H
That is.
1/2 2/3 3/4
blzal b2:a2 b3:a3
Prove that 2 bn must converge. Hint: find an upper bound for the partial
sums.
