5.33. In Example 5.37 we used a differentiation trick to compute the value of the infinite series 1 np(1-p)n-1. This exercise further develops this useful technique. The starting point is the formula for the geometric series n=0 and the differential operator x = 1 - X d dx (a) Using the fact that D(x") = nx", prove that D = x- Σnx" n=1 Σn²x¹ n=0 for x < 1 = [³ n=0 by applying D to both sides of (5.57). For which does the left-hand side of (5.58) converge? (Hint. Use the ratio test.) (b) Applying D again, prove that x (1-x)² nkx" = x+x² (1-x)³ (5.57) (5.58) (c) More generally, prove that for every value of k there is a polynomial F(x) such that Fk (x) (1-x)k+¹* (5.59) (5.60) (Hint. Use induction on k.) (d) The first few polynomials Fk (x) in (c) are Fo(x) = 1, F₁(x) = x, and F2(x) = x+x². These follow from (5.57), (5.58), and (5.59). Compute F3(x) and F₁(x).
5.33. In Example 5.37 we used a differentiation trick to compute the value of the infinite series 1 np(1-p)n-1. This exercise further develops this useful technique. The starting point is the formula for the geometric series n=0 and the differential operator x = 1 - X d dx (a) Using the fact that D(x") = nx", prove that D = x- Σnx" n=1 Σn²x¹ n=0 for x < 1 = [³ n=0 by applying D to both sides of (5.57). For which does the left-hand side of (5.58) converge? (Hint. Use the ratio test.) (b) Applying D again, prove that x (1-x)² nkx" = x+x² (1-x)³ (5.57) (5.58) (c) More generally, prove that for every value of k there is a polynomial F(x) such that Fk (x) (1-x)k+¹* (5.59) (5.60) (Hint. Use induction on k.) (d) The first few polynomials Fk (x) in (c) are Fo(x) = 1, F₁(x) = x, and F2(x) = x+x². These follow from (5.57), (5.58), and (5.59). Compute F3(x) and F₁(x).
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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Transcribed Image Text:5.33. In Example 5.37 we used a differentiation trick to compute the value of the
infinite series ₁ np(1-p)n-¹. This exercise further develops this useful technique.
The starting point is the formula for the geometric series
n=0
and the differential operator
x"
n
=
1
2, for x < 1
- x
D = x
Σnx"
n=1
(a) Using the fact that D(x") = nx", prove that
d
2
Σn²x"
Ση
n=0
n=0
dx
(5.58)
by applying D to both sides of (5.57). For which x does the left-hand side
of (5.58) converge? (Hint. Use the ratio test.)
(b) Applying D again, prove that
=
=
x
(1 - x)²
=
x+x²
(1-x)³*
(5.57)
(c) More generally, prove that for every value of k there is a polynomial F(x) such
that
Fk (x)
(1-x)k+1*
(5.59)
(5.60)
(Hint. Use induction on k.)
(d) The first few polynomials Fk (x) in (c) are Fo(x) = 1, F₁(x) = x, and F₂(x) =
x+x². These follow from (5.57), (5.58), and (5.59). Compute F3(x) and F₁(x).
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