Task #4: Alternating Series Test
Task #4: Alternating Series Test
∑
j=1
∞
(−1)j1
√
(j+c)
c= 2*birth month +5=2*6+5=17
∑
j=1
∞
(−1)j1
√
(j+17)
A) State thethreeconditions of the alternating series test.
1. The
Un
's
are all positive
2. The positive
Un
's
are (eventually) Nonincreasing
Un
≥ U n+1
for all n
≥ N
, for some
integer N.
3.
Un→0
B) Justify thateachcondition of the alternating series test from part A is true for the infinite
series in the Assumptions section, using algebra or calculus or both.
1. The
Un
's
are all positive
-The numerator is positive 1 and so it will stay positive.
-The denominator is
√
j+17
and will always be greater than zero when j
≥1
. So the
denominator will positive. Since the numerator is positive and the denominator is positive
the
Un
's
are all positive.
2. We must determine if the function is nonincreaing and to do that we must show that
the derivative is negative.
f
(
j
)
=1
√
j+17
¿1
(
j+17
)
(1
2)
=
(
j+17
)
(−1
2)
let u=(j+17)
Apply the chain rule:
d
du
(
u
−1
2
)
(
d
dj
)
(j+17)
Apply the power rule and simplify:
(u¿¿−1
2−1)=−1
2u
−3
2¿
=
−1
2u
3
2
Apply the sum rule
d
dj
(
j+17
)
=1
