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3 Questions on Geometric Series - Calculus II - Assignment | MATH 206, Assignments of Calculus

Material Type: Assignment; Class: Calculus II; University: University of Hawaii at Hilo; Term: Unknown 1989;

Typology: Assignments

2009/2010

Uploaded on 04/12/2010

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Math 206 - Calculus II
Homework due October 23
Recall that in class we developed the Geometric Series. If |r|<1, then we
have that
r0+r1+r2+. . . =
X
k=0
rk=1
1r.
Question 1. Use the Geometric Series to compute the following infinite
sums. Be sure to note the starting kvalue of these series.
(a)
X
k=0 3
4k
(b)
X
k=0
9
4k
(c)
X
k=2
(1)k1
2k
(d)
X
k=0
2k+1
7k
(e)
X
k=0 e
πk
(f)
X
k=0
e
πne
Question 2. Can a Geometric Series
X
k=0
rk
ever converge to a negative number? If so, give an example. If not, why not?
Can one every converge to 0? If so, give an example. If not, why not?
Question 3. Use the Geometric Series to write the following decimals as
fractions.
(a) 3.77777 . . .
(b) 1.414 = 1.414414414414 . . .
1

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Math 206 - Calculus II

Homework due October 23

Recall that in class we developed the Geometric Series. If |r| < 1, then we have that

r^0 + r^1 + r^2 +... =

∑^ ∞

k=

rk^ =

1 − r

Question 1. Use the Geometric Series to compute the following infinite sums. Be sure to note the starting k value of these series.

(a)

∑^ ∞

k=

)k

(b)

∑^ ∞

k=

4 k

(c)

∑^ ∞

k=

(−1)k^

2 k

(d)

∑^ ∞

k=

2 k+ 7 k

(e)

∑^ ∞

k=

( (^) e π

)k

(f)

∑^ ∞

k=

enπ πne

Question 2. Can a Geometric Series

∑^ ∞

k=

rk

ever converge to a negative number? If so, give an example. If not, why not? Can one every converge to 0? If so, give an example. If not, why not?

Question 3. Use the Geometric Series to write the following decimals as fractions.

(a) 3. 77777...

(b) 1.414 = 1. 414414414414...