The following alternating series converge to given multiples of π . Find the value of N predicted by the remainder estimate such that the Nth partial sum of the series accurately approximates the left-hand side to within the given error. Find the minimum N for which the error bound holds, and give the desired approximate value in each case. Up to 15 decimals places, π =3.141592653589793… 308. [T] The series ∑ n = 0 ∞ sin ( x + π n ) x + π n plays an important role in signal processing. Show that E ∑ n = 0 ∞ sin ( x + π n ) x + π n converges whenever 0 π . (Hint: Use the formula for the sum of a sum of angles.)
The following alternating series converge to given multiples of π . Find the value of N predicted by the remainder estimate such that the Nth partial sum of the series accurately approximates the left-hand side to within the given error. Find the minimum N for which the error bound holds, and give the desired approximate value in each case. Up to 15 decimals places, π =3.141592653589793… 308. [T] The series ∑ n = 0 ∞ sin ( x + π n ) x + π n plays an important role in signal processing. Show that E ∑ n = 0 ∞ sin ( x + π n ) x + π n converges whenever 0 π . (Hint: Use the formula for the sum of a sum of angles.)
The following alternating series converge to given multiples of
π
. Find the value of N predicted by the remainder estimate such that the Nth partial sum of the series accurately approximates the left-hand side to within the given error. Find the minimum N for which the error bound holds, and give the desired approximate value in
each case. Up to 15 decimals places,
π
=3.141592653589793…
308. [T] The series
∑
n
=
0
∞
sin
(
x
+
π
n
)
x
+
π
n
plays an important role in signal processing. Show that E
∑
n
=
0
∞
sin
(
x
+
π
n
)
x
+
π
n
converges whenever 0
π
. (Hint: Use the formula for the sum
of a sum of angles.)
University Calculus: Early Transcendentals (4th Edition)
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