Consider the interval [-r, r], and suppose f: I → R is an infinitely differentiable function. Let Pn be the n-th Maclaurin polynomial, and let Rn(x) = f(x) – Pn(x) be the remainder. Mrn+1 (a) Suppose |f(n+1)(x)| < M for all x € [-r, r]. Show that |R,(x)| < for all r E [-r, r]. (n + 1)! (Hint: Use Theorem 8.2.4.) e"rn+1 (b) Show that e – Σ k! k=0 (n + 1)! for all æ E [-r, r]. (c) Given that e E [2,3], determine how many terms of the series E are required to compute the value of e to an accuracy of 5 decimal places. (Hint: Use part (b) with an appropriate r.)
Consider the interval [-r, r], and suppose f: I → R is an infinitely differentiable function. Let Pn be the n-th Maclaurin polynomial, and let Rn(x) = f(x) – Pn(x) be the remainder. Mrn+1 (a) Suppose |f(n+1)(x)| < M for all x € [-r, r]. Show that |R,(x)| < for all r E [-r, r]. (n + 1)! (Hint: Use Theorem 8.2.4.) e"rn+1 (b) Show that e – Σ k! k=0 (n + 1)! for all æ E [-r, r]. (c) Given that e E [2,3], determine how many terms of the series E are required to compute the value of e to an accuracy of 5 decimal places. (Hint: Use part (b) with an appropriate r.)
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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![3. Consider the interval [-r, r], and suppose f: I → R is an infinitely differentiable function. Let pn
be the n-th Maclaurin polynomial, and let Rn(x) = f(x) – Pn (x) be the remainder.
Mrn+1
(a) Suppose |f(n+1) (x)| < M for all x € [-r,r]. Show that |R,(x)| <
for all æ E [-r, r].
(n +1)!
(Hint: Use Theorem 8.2.4.)
e"rn+1
(b) Show that Je*
Σ
for all x € [-r, r].
k!
k=0
(n + 1)!
(c) Given that e E [2,3], determine how many terms of the series E are required to compute
the value of e to an accuracy of 5 decimal places.
(Hint: Use part (b) with an appropriate r.)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fab68ef6f-9760-4c1f-88a8-6e5ceb2168b2%2F9dbf0ac2-1721-4092-b0f6-56f8a68430a6%2Fqrfr5vr_processed.jpeg&w=3840&q=75)
Transcribed Image Text:3. Consider the interval [-r, r], and suppose f: I → R is an infinitely differentiable function. Let pn
be the n-th Maclaurin polynomial, and let Rn(x) = f(x) – Pn (x) be the remainder.
Mrn+1
(a) Suppose |f(n+1) (x)| < M for all x € [-r,r]. Show that |R,(x)| <
for all æ E [-r, r].
(n +1)!
(Hint: Use Theorem 8.2.4.)
e"rn+1
(b) Show that Je*
Σ
for all x € [-r, r].
k!
k=0
(n + 1)!
(c) Given that e E [2,3], determine how many terms of the series E are required to compute
the value of e to an accuracy of 5 decimal places.
(Hint: Use part (b) with an appropriate r.)
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