The following series do not satisfy the hypotheses of the alternating series test as stated. In each case, state which hypothesis is not satisfied. State whether the series converges absolutely. 305. Sometimes the alternating series ∑ n = ∞ ( − 1 ) n − 1 b n converges to a certain fraction of an absolutely convergent series ∑ n = 1 ∞ b n at a faster rate. Given that ∑ n = 1 ∞ 1 n 2 − π 2 6 , find S = 1 − 1 2 2 + 1 3 2 + 1 4 2 + ... Which of the series 6 ∑ n = 1 ∞ 1 n 2 and S ∑ n = 1 ∞ ( − 1 ) n − 1 n 2 gives a better estimation of π 2 using 1000 terms?
The following series do not satisfy the hypotheses of the alternating series test as stated. In each case, state which hypothesis is not satisfied. State whether the series converges absolutely. 305. Sometimes the alternating series ∑ n = ∞ ( − 1 ) n − 1 b n converges to a certain fraction of an absolutely convergent series ∑ n = 1 ∞ b n at a faster rate. Given that ∑ n = 1 ∞ 1 n 2 − π 2 6 , find S = 1 − 1 2 2 + 1 3 2 + 1 4 2 + ... Which of the series 6 ∑ n = 1 ∞ 1 n 2 and S ∑ n = 1 ∞ ( − 1 ) n − 1 n 2 gives a better estimation of π 2 using 1000 terms?
The following series do not satisfy the hypotheses of the alternating series test as stated.
In each case, state which hypothesis is not satisfied. State whether the series converges absolutely.
305. Sometimes the alternating series
∑
n
=
∞
(
−
1
)
n
−
1
b
n
converges to a certain fraction of an absolutely convergent series
∑
n
=
1
∞
b
n
at a faster rate. Given that
∑
n
=
1
∞
1
n
2
−
π
2
6
, find
S
=
1
−
1
2
2
+
1
3
2
+
1
4
2
+
...
Which of the series
6
∑
n
=
1
∞
1
n
2
and
S
∑
n
=
1
∞
(
−
1
)
n
−
1
n
2
gives a better estimation
Using and Understanding Mathematics: A Quantitative Reasoning Approach (6th Edition)
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