In the following exercises, using a substitution if indicated, express each series in terms of elementary functions and find the radius of convergence of the sum. 114. [T] Given the power series expansion tan − 1 ( x ) = ∑ k = 0 ∞ ( − 1 ) k x 2 k + 1 2 k + 1 , use the alternating series test to determine how many terms N of the sum evaluated at x = 1 are needed to approximate tan-1(1) = π 4 accurate to within 1/1000. Evaluate the corresponding partial stun ∑ k = 0 ∞ ( − 1 ) k x 2 k + 1 2 k + 1 .
In the following exercises, using a substitution if indicated, express each series in terms of elementary functions and find the radius of convergence of the sum. 114. [T] Given the power series expansion tan − 1 ( x ) = ∑ k = 0 ∞ ( − 1 ) k x 2 k + 1 2 k + 1 , use the alternating series test to determine how many terms N of the sum evaluated at x = 1 are needed to approximate tan-1(1) = π 4 accurate to within 1/1000. Evaluate the corresponding partial stun ∑ k = 0 ∞ ( − 1 ) k x 2 k + 1 2 k + 1 .
In the following exercises, using a substitution if indicated, express each series in terms of elementary functions and find the radius of convergence of the sum.
114. [T] Given the power series expansion
tan
−
1
(
x
)
=
∑
k
=
0
∞
(
−
1
)
k
x
2
k
+
1
2
k
+
1
, use the alternating series test to determine how many terms N of the sum evaluated at
x = 1 are needed to approximate tan-1(1) =
π
4
accurate to within 1/1000. Evaluate the corresponding partial stun
∑
k
=
0
∞
(
−
1
)
k
x
2
k
+
1
2
k
+
1
.
Using & Understanding Mathematics: A Quantitative Reasoning Approach (7th Edition)
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