Math 118 Spring 2023: Practice Set 6
This assignment is for practice only. It is not to be submitted.
1. Write the number 2.317 = 2.3171717... as a ratio of integers.
Hint: Treat 0.0171717... as an infinite sum 0.017 + 0.00017 + 0.0000017 + ....
2. In each case, determine if the given series is convergent or divergent. If it is convergent,
determine the sum.
(a)
(b)
(c)
(d)
∞
X
n=1
∞
X
n=1
∞
X
n=1
∞
X
n=1
22n 31−n
22n 51−n
21/n
3n
2n + 7
3. When light hits a certain pane of glass, the glass reflects one half of the light, absorbs
one fourth, and transmits one fourth. A window is made of two panes of this glass
separated by a small gap. If light of intensity I shines directly onto the window, what
fraction is transmitted to the other side of the double pane?
4. Find a simple expression for the nth partial sum of the series
∞ X
k=2
2
2
k −1
and hence find its sum.
Hint: Use partial fractions. This type of series is referred to as “telescoping” because
an expression with many terms collapses into a simpler expression.
5. Use the integral test to determine whether the series converges or diverges:
(a)
(b)
∞
X
4
n=1
∞
X
n=2
(c)
1
(n + 1) 3
1
p
n ln(n)
∞
X
21/n
n=1
n2
6. Use the comparison test to determine whether the series converges or diverges:
(a)
∞
X
n=1
∞
X
1
√
n+ n
1
√
3
n +1
n=1
√
∞
X
n2 + 1 arctan(n)
(c)
n3
n=1
(b)
7. Use the limit comparison test to determine whether the series converges or diverges:
√
∞
X
n
(a)
n4 + n
n=1
(b)
∞
X
n=2
1
n − ln(n)