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. 61. [T] Plot the graphs of the partial sums S N = ∑ n = 0 N ( − 1 ) n x 2 n + 1 ( 2 n + 1 ) ! For n =3, 5, 10 on the interval [ − 2 π , 2 π ] . Comment on the how these plots approximate sin x 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. 61. [T] Plot the graphs of the partial sums S N = ∑ n = 0 N ( − 1 ) n x 2 n + 1 ( 2 n + 1 ) ! For n =3, 5, 10 on the interval [ − 2 π , 2 π ] . Comment on the how these plots approximate sin x 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.
61. [T] Plot the graphs of the partial sums
S
N
=
∑
n
=
0
N
(
−
1
)
n
x
2
n
+
1
(
2
n
+
1
)
!
For n =3, 5, 10 on the interval
[
−
2
π
,
2
π
]
. Comment on the how these plots approximate
sin
x
as N increases.
Calculus for Business, Economics, Life Sciences, and Social Sciences (14th Edition)
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