Question 3. Let f(x) (sin x)2 for x ER and let [x] denote the largest integer less than or equal to x. = (a) State Taylor's Theorem. (b) Prove by mathematical induction on n that for n ≥ 1, f(n)(x) = (-1)"¹12"-sin 2x (-1) ¹2n-1 cos 2x if n is odd if n is even. (c) Write down the Taylor series for f(x) about x = 0. (d) Determine for which x R the Taylor series for f(x) equals f(x).
Question 3. Let f(x) (sin x)2 for x ER and let [x] denote the largest integer less than or equal to x. = (a) State Taylor's Theorem. (b) Prove by mathematical induction on n that for n ≥ 1, f(n)(x) = (-1)"¹12"-sin 2x (-1) ¹2n-1 cos 2x if n is odd if n is even. (c) Write down the Taylor series for f(x) about x = 0. (d) Determine for which x R the Taylor series for f(x) equals f(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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![Question 3. Let f(x) = (sin x)² for x = R and let [x] denote the largest
integer less than or equal to x.
(a) State Taylor's Theorem.
(b) Prove by mathematical induction on n that for n ≥ 1,
f(n)(x) =
(-1)"¹12"-sin 2x
(−1)[¹]2n-¹ cos 2x
if n is odd
if n is even.
(c) Write down the Taylor series for f(x) about x = 0.
(d) Determine for which x € R the Taylor series for f(x) equals f(x).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffef45238-eb5b-4e87-99b3-bef83613c2a6%2F741bb365-9ed3-46bd-97b4-a2882198c469%2Fm5gp8v_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Question 3. Let f(x) = (sin x)² for x = R and let [x] denote the largest
integer less than or equal to x.
(a) State Taylor's Theorem.
(b) Prove by mathematical induction on n that for n ≥ 1,
f(n)(x) =
(-1)"¹12"-sin 2x
(−1)[¹]2n-¹ cos 2x
if n is odd
if n is even.
(c) Write down the Taylor series for f(x) about x = 0.
(d) Determine for which x € R the Taylor series for f(x) equals f(x).
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