Chapte 434 1. (1+x)'/2 = 1 +* (c) 2. 2.4 1.3 x'+... 2.4.6 1.3.5...(2n – 3) + (-1)"+1 2.4.6...(2n) +……. .... 1 (1+x)-1/2 = 1- (d) +... 2.4 1-3.5...(2n – 1) +(-1)" x"+... 2.4.6...(2n) 9. Prove that the inequality 2 < (1+ 1/n)" < 3 holds for all integers n > 1. [Hint: By the binomial theorem, (1+)-1+ 1 < 1 +1+ 1 +. 2.3 1 +.. 2.3.4...n 1 < 1 + 1. 1 +... 22 1 2n-1*
Chapte 434 1. (1+x)'/2 = 1 +* (c) 2. 2.4 1.3 x'+... 2.4.6 1.3.5...(2n – 3) + (-1)"+1 2.4.6...(2n) +……. .... 1 (1+x)-1/2 = 1- (d) +... 2.4 1-3.5...(2n – 1) +(-1)" x"+... 2.4.6...(2n) 9. Prove that the inequality 2 < (1+ 1/n)" < 3 holds for all integers n > 1. [Hint: By the binomial theorem, (1+)-1+ 1 < 1 +1+ 1 +. 2.3 1 +.. 2.3.4...n 1 < 1 + 1. 1 +... 22 1 2n-1*
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
8c: use the binomial theorem to obtain the following series expansions
Transcribed Image Text:Chapte
434
1.
(1+x)'/2 = 1 +*
(c)
2.
2.4
1.3
x'+...
2.4.6
1.3.5...(2n – 3)
+ (-1)"+1
2.4.6...(2n)
+…….
....
1
(1+x)-1/2 = 1-
(d)
+...
2.4
1-3.5...(2n – 1)
+(-1)"
x"+...
2.4.6...(2n)
9. Prove that the inequality 2 < (1+ 1/n)" < 3 holds for
all integers n > 1. [Hint: By the binomial theorem,
(1+)-1+
1
< 1 +1+
1
+.
2.3
1
+..
2.3.4...n
1
< 1 + 1.
1
+...
22
1
2n-1*
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