Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
3d
![9:26 MEI ✔
C
(d)
(e)
(c)
(1) + ² ( 2 ) + ³ ( 3 ) + ··· + n(") = n2²−¹.
3
n2n-1
(f)
(b)
(6) + (0+3)
(c)
5
each of the identities al 28%
[Hint: Let a = b = 1 in the binomial theorem.]
-- + (−1)" ("")
[Hint: After expanding n(1 + b)"-1 by the binomial theore
that
(0) + ² (7) + ²² (²)
(8) + (2) + (4) + (3)
(1) + (3) + (3)
=
4. Prove the following for n ≥ 1:
n (" z ¹)
k
[Hint: Use parts (a) and (b).]
1
(0) − ½ (01) + ½ (2) -
-
3
[Hint: The left-hand side equals
1
|||
5. (a)en 22, pe that
+... + 2"
= (k+1)
+...
O
„+₁[("†')-("+¹)+("‡¹) ---
2
n
¹)(1+1).J
k+
(3)
24 of 450
+ ... = 2n-1.
(-1)"
n+1
n
1
(a)
(") <(₁ + ₁) if and only if 0 < r < (n − 1).
r+1
n
1)
if and only if n - 1 ≥ r > −(n − 1).
(n-1
r+1
(7) > ( , + ₁)
(7) - (, 4₁)
n
=
= 3".
(1)=+++
n
0
3
if and only if n is an odd integer, and r =
+(-
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Transcribed Image Text:9:26 MEI ✔
C
(d)
(e)
(c)
(1) + ² ( 2 ) + ³ ( 3 ) + ··· + n(") = n2²−¹.
3
n2n-1
(f)
(b)
(6) + (0+3)
(c)
5
each of the identities al 28%
[Hint: Let a = b = 1 in the binomial theorem.]
-- + (−1)" ("")
[Hint: After expanding n(1 + b)"-1 by the binomial theore
that
(0) + ² (7) + ²² (²)
(8) + (2) + (4) + (3)
(1) + (3) + (3)
=
4. Prove the following for n ≥ 1:
n (" z ¹)
k
[Hint: Use parts (a) and (b).]
1
(0) − ½ (01) + ½ (2) -
-
3
[Hint: The left-hand side equals
1
|||
5. (a)en 22, pe that
+... + 2"
= (k+1)
+...
O
„+₁[("†')-("+¹)+("‡¹) ---
2
n
¹)(1+1).J
k+
(3)
24 of 450
+ ... = 2n-1.
(-1)"
n+1
n
1
(a)
(") <(₁ + ₁) if and only if 0 < r < (n − 1).
r+1
n
1)
if and only if n - 1 ≥ r > −(n − 1).
(n-1
r+1
(7) > ( , + ₁)
(7) - (, 4₁)
n
=
= 3".
(1)=+++
n
0
3
if and only if n is an odd integer, and r =
+(-
:
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