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.