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Assignment 5 Remarks and Partial Solutions
14.1 Determine which of the following series converge. Justify your answers.
(a)
∑
n
4
2
n
. (Hueristics: The exponential is the
dominant
term and will overcome any power.
Therefore, we expect this to behave more like
∑
1
2
n
(although it never quite achieves
this). For this combination of powers and exponential, it is equally advantageous to use
the Ratio or Root Test.)
limsup
±
±
±
±
±
²
n
4
2
n
³
1
/n
±
±
±
±
±
= limsup
(
n
1
/n
)
4
2
!
=
1
2
.
Therefore, it converges absolutely by the root test. (Note: The root test for the series
∑
1
2
n
also gives
1
2
.)
(b)
∑
2
n
n
!
. (Hueristics: You probably don’t have much intuition about the factorial function
n
!. It turns out that it behaves a little faster in growth than
n
n
e

n
by a factor like
√
n
(something called Stirling’s formula). You will see this when you apply the Ratio Test
since the factor
n
n
will drive the ratio to zero. Whenever you encounter the factorial
function in a series, the ratio test is a natural ﬁrst try.)
limsup
±
±
±
±
a
n
+1
a
n
±
±
±
±
= limsup
²
2
n
+ 1
³
= 0
.
Therefore, it converges absolutely by the ratio test.
(c)
∑
n
2
3
n
. Similarly to (a), it converges absolutely by the root test.
(d)
∑
n
!
3
n
. Will diverge by the ratio test (see the hueristics of (b)).
(e)
∑
cos
2
(
n
)
n
2
. (Hueristics: The cosine function is bounded

cos(
x
)
 ≤
1. Thus, one should
see what else is there. In this case it is
1
n
2
which we know to sum nicely. Hence try to
compare it to
∑
1
n
2
.)
Since
a
n
=
cos
2
(
n
)
n
2
≤
1
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 Spring '09

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