1. Let c > 0 be fixed and define the sequence {an} by setting an (an? + 3c) а1 > 0, аn+1 for n E N. Зап? + с 3a,? -
1. Let c > 0 be fixed and define the sequence {an} by setting an (an? + 3c) а1 > 0, аn+1 for n E N. Зап? + с 3a,? -
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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![1. Let c > 0 be fixed and define the sequence {an} by setting
an (an? + 3c)
За,2 + с
2
а1 > 0,
An+1
for n E N.
Determine all a1 for which the sequence converges and in such a case
find its limit.
2. Prove that if f is a bounded function on [0,1] satisfying f(ax)
bf (x) for 0 < x <- and a, b > 1, then lim f(x) = f(0).
| 1 3. Suppose that f:[1, c0) → R is uniformly continuous. Prove that
there is a positive M such that-
If (x)|
< M for x > 1.
4. A function f: [0,1] → R satisfies f (0) < 0 and f(1) > 0, and there
exists a function g continuous on [0,1] and such that f + g is decreasing.
Prove that the equation f (x) = O has a solution in the open interval (0,1).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fca76f848-e5e0-4b90-aa1a-531febb1197d%2Ff59331ee-68be-48a1-83ab-7076f650c599%2Fymrfh9b_processed.jpeg&w=3840&q=75)
Transcribed Image Text:1. Let c > 0 be fixed and define the sequence {an} by setting
an (an? + 3c)
За,2 + с
2
а1 > 0,
An+1
for n E N.
Determine all a1 for which the sequence converges and in such a case
find its limit.
2. Prove that if f is a bounded function on [0,1] satisfying f(ax)
bf (x) for 0 < x <- and a, b > 1, then lim f(x) = f(0).
| 1 3. Suppose that f:[1, c0) → R is uniformly continuous. Prove that
there is a positive M such that-
If (x)|
< M for x > 1.
4. A function f: [0,1] → R satisfies f (0) < 0 and f(1) > 0, and there
exists a function g continuous on [0,1] and such that f + g is decreasing.
Prove that the equation f (x) = O has a solution in the open interval (0,1).
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