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