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Mathematics
Mathematics
Problem
Using the first four nonzero terms of the Maclaurin series
arctan
x
=
∑
n
=
0
∞
(
−
1
)
n
x
2
n
+
1
2
n
+
1
\displaystyle\arctan x =\sum_{n=0}^{\infty}(-1)^{n}\dfrac{x^{2n+1}}{2n+1}
arctan
x
=
n
=
0
∑
∞
(
−
1
)
n
2
n
+
1
x
2
n
+
1
, form an interval that is guaranteed to contain
π
4
\dfrac{\pi}{4}
4
π
. Express both bounds as exact rational numbers.
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