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Unformatted text preview: rsity X account.
Payment Schedule
A. The amount of payment on the payment # n is exactly
1
.
one nth of a dollar
=$
n
See Table 1. B. As for the timing of payments, the interval between two consecutive payments
is exactly one half of the previous interval. See Table 2.
C. The payment will conclude 48 hours after the payment # 1, but not before.
payment # amount #1 $1 #2 $ 1 /2 #3 $ 1 /3 #4 $ 1 /4 #5 $ 1 /5 #6 $ 1 /6 .
.
. .
.
.
Table 1.
2 payment # timing of payment #1 −− #2 24 hours after payment #1 #3 24/2 hours after payment #2 #4 24/4 hours after payment #3 #5 24/8 hours after payment #4 #6 24/16 hours after payment #5 #7 24/32 hours after payment #6 .
.
. .
.
.
Table 2. Suppose you are Madam Rich’s ﬁnancial advisor. Your job is not to let her lose all
her money. Should you tell her that it is okay to sign? Explain. 3 [III] (Takehome; 20pts) fx gx (1a) = = −1
2 For x with −2 < x < 2, deﬁne
∞ ln
n=2 x2
n2 1− n , 1 x+
lim − k
n− ∞
→
k=2 n 1
2 k+x .
k−x ln
k=2 Agree that f x can be written as −1
lim ln 2
n− ∞
→ 1+ x
2 x
2 1− x
3 1+ ··· · 1+ x
n 1− 1− x
n x
3 · . Agreed.
(1b) Agree that g x can be written as n 1
1
lim − x+
ln
k
2
n− ∞
→
k=2 1+ x
2 1− x
2 Agreed.
4 1+ x
3 · ··· · 1 + x
n 1− x
3 · ··· · 1 − x
n . (2a) Give the Taylor series expression of
−f x Answer for (2a) ∞ m=2 −1 . : ζm −1
m m−1 −1 x −1 ζ
+ x xm + − ζ
=− −1 ζ x − −1 ζ x −1 + ··· . ζ x : Work for (2a) −f x +gx +gx = lim − n− ∞
→ n k=2 =
lim − n− ∞
→ n
k=2 1
x
x + ln 1 +
k
2 1
x+
k n ln 1+ k=2 5 1+ x
k x
3 · ··· · 1+ x
n = n 1
lim − x+
k
n− ∞
→
k=2 n = lim
n− ∞
→ k=2 n = lim
n− ∞
→ k=2 ∞ =
k=2 − − −1 x
k x
k −1 = k=2 + −1 ∞ + −1 +
m=2 m=2 −1 ∞ =
m=2 −1 m−1 m−1 x
k x
k m−1 x
k two Σs were interchanged k=2 x m−1 m−1 k=2 ∞ x
k ∞ = x
k m−1 m=1 x
k m−1 m=1 −1 ∞ m=2 ∞ m=2 n...
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 Fall '08
 SCHOLLE,MINHO
 Calculus

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