95. The following "post-Newtonian" approximation of a uses the “Gregory Series":x+5xHere is an informal derivation of the series: tan¯X = x+371(a) Use the substitution t = tan a to show "-1dt = tanx.21(b) Now expand= (1+1²)* by the binomial series and integrate the series term-wise1+t?from 0 to x, and thereby derive the Gregory series.

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95. The following “post-Newtonian" approximation of t uses the "Gregory Series": Here is an informal derivation of the series: tan -1 X = X -- + 3 5 +... 7 (a) Use the substitution t = tan a to show J. 2 1 dt = tan-x. (b) Now expand 1 (1+t)* by the binomial series and integrate the series term-wise 1+t? from 0 to x, and thereby derive the Gregory series.

95. The following “post-Newtonian" approximation of t uses the "Gregory Series": Here is an informal derivation of the series: tan -1 X = X -- + 3 5 +... 7 (a) Use the substitution t = tan a to show J. 2 1 dt = tan-x. (b) Now expand 1 (1+t)* by the binomial series and integrate the series term-wise 1+t? from 0 to x, and thereby derive the Gregory series.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

95. The following "post-Newtonian" approximation of a uses the “Gregory Series":
x
+
5
x
Here is an informal derivation of the series: tan¯
X = x
+
3
7
1
(a) Use the substitution t = tan a to show "
-1
dt = tanx.
2
1
(b) Now expand
= (1+1²)* by the binomial series and integrate the series term-wise
1+t?
from 0 to x, and thereby derive the Gregory series.

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