In the following exercises, suppose that p ( x ) = ∑ n = 0 ∞ a n x n Satisfies lim n → ∞ a n + 1 a n = 1 where a n ≥ 0 for each n . State whether each series converges on the full interval (− 1, 1), or if there is not enough information to draw a conclusion. Use the comparison test when appropriate. 57. [T] Plot the graphs of 1 1 − x and of the partial sums S N = ∑ n = 0 N x n for n = 10 , 20 , 30 on the interval [-0.99, 0.99]. Comment on the approximation of 1 1 − x by S N near x = − 1 and near x = 1 as N increases.
In the following exercises, suppose that p ( x ) = ∑ n = 0 ∞ a n x n Satisfies lim n → ∞ a n + 1 a n = 1 where a n ≥ 0 for each n . State whether each series converges on the full interval (− 1, 1), or if there is not enough information to draw a conclusion. Use the comparison test when appropriate. 57. [T] Plot the graphs of 1 1 − x and of the partial sums S N = ∑ n = 0 N x n for n = 10 , 20 , 30 on the interval [-0.99, 0.99]. Comment on the approximation of 1 1 − x by S N near x = − 1 and near x = 1 as N increases.
In the following exercises, suppose that
p
(
x
)
=
∑
n
=
0
∞
a
n
x
n
Satisfies
lim
n
→
∞
a
n
+
1
a
n
=
1
where
a
n
≥
0
for each
n
. State whether each series converges on the full interval (− 1, 1), or if there is not enough information to draw a conclusion. Use the comparison test when appropriate.
57. [T] Plot the graphs of
1
1
−
x
and of the partial sums
S
N
=
∑
n
=
0
N
x
n
for
n
=
10
,
20
,
30
on the interval [-0.99, 0.99]. Comment on the approximation of
1
1
−
x
by
S
N
near
x
=
−
1
and near
x
=
1
as N increases.
Calculus for Business, Economics, Life Sciences, and Social Sciences (14th Edition)
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