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
1−r.
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
enπ
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