Exam (elaborations)
Taylor and Maclaurin Series solved questions
- Course
- Institution
Taylor and Maclaurin Series solved questions
[Show more]
Seller
Follow
jureloqoo
Content preview
CHAPTER 39Taylor and Maclaurin Series
39.1 Find the Maclaurin series of e*.
Let /(*) = **. Then f("\x) = e' for all «>0. Hence, /'"'(O) = 1 for all n & 0 . Therefore, the
Maclaurin series
39.2 Find the Maclaurin series for sin x.
Let /(;c) = sin X. Then, /(0) = sinO = 0, /'(O) = cosO= 1, /"(O) = -sin 0 = 0, /"'(O) = -cosO= -1,
and, thereafter, the sequence of values 0,1,0, — 1 keeps repeating. Thus, we obtain
39.3 Find the Taylor series for sin x about ir/4.
Let f(x) = sinx. Then /(ir/4) = sin (ir/4) = V2/2, f ' ( i r / 4 ) = cos(irf4) = V2/2, f(ir/4) = -sin (77/4) =
and, thereafter, this cycle of four values keeps repeating.
Thus, the Taylor series for sin* about
39.4 Calculate the Taylor series for IIx about 1.
Let Then, and, in general,
So /(">(1) = (-!)"«!. Thus, the Taylor series is
39.5 Find the Maclaurin series for In (1 - x).
Let/(AT) = In (1 - x). Then,/(0) = 0,/'(0) = -1,/"(0) = -1,/"'(0) = -1 -2,/ < 4 ) = -1 • 2 • 3, and, in general
/ ( ">(0) = -(„ _ i)i Thus, for n > l,/ ( / 0 (0)/n! = -1/n, and the Maclaurin series is
39.6 Find the Taylor series for In x around 2.
Let f(x) = lnx. Then, and, in general,
So /(2) = In 2, and, for n > 1, Thus, the
Taylor series is (x - 2)" = In 2 + i(x - 2) - 4(x - 2)2 + MX - 2)3 - • • •.
39.7 Compute the first three nonzero terms of the Maclaurin series for ecos".
Let f ( x ) = ecos*. Then, f'(x)=-ecos'sinx, /"(*) = e'05* (sin2 *-cos*), /'"(x) = ecos' (sin jc)(3 cos x +
1-sin 2 *), /(4)(;<:) = e<:osj:[(-sin2*)(3 + 2cosA-) + (3cosA: + l-sin 2 Ar)(cosA:-sin 2 jc)]. Thus, /(O) = e,
/'(0) = 0, f"(0) = -e, f"(0) = 0, /(4)(0) = 4e. Hence, the Maclaurin series is e(l - |*2 + |x4 + • • • ) .
340
, TAYLOR AND MACLAURIN SERIES 341
39.8 Write the first nonzero terms of the Maclaurin series for sec x.
I Let /(x) = secx. Then, /'(*) = sec x tan x, f"(x) = (sec *)(! + 2 tan 2 x), /'"(*) = (sec x tan x)(5 + 6 tan 2 x),
fw(x) = 12 sec3 x tan 2 x + (5 + 6 tan2 x)(sec3 x + tan 2 x sec x). Thus, /(O) = 1, /'(O) = 0, /"(O) = 1,
/"'(0) = 0, / ( 4 ) =5. The Maclaurin series is 1+ $x2 + &x* + • • •.
39.9 Find the first three nonzero terms of the Maclaurin series for tan x.
Let /(x) = tan;t. Then /'(*) = sec2*, /"(*) = 2 tan x sec2 x, f'"(x) = 2(sec4 x + 2 tan 2 x + 2 tan 4 x),
/ (*) = 8 (tan x sec2 x)(2 + 3 tan2 x),
(4)
/(5)(*) = 48 tan 2 x sec4 x + 8(2 + 3 tan2 *)(sec4 x + 2 tan 2 x + 2 tan 4 x).
So /(0) = 0, /'(0) = 1, f'(0) = 0, /"'(0) = 2, /< 4 >(0) = 0, / <5) (0) = 16. Thus, the Maclaurin series is
*+ 1*3 + &X3 + ••-.
39.10 Write the first three nonzero terms of the Maclaurin series for sin -1x.
Let /Ot^sirT1*. /'(*) = (I-*2)'1'2, /"(x) = *(1 - Jc2)'3'2, f'"(x) = (1 - x2ys'2(2x2 + 1).
/(4)(*) = 3*(l-*2)-7/2(2*2 + 3), /(5)(*) = 3(1 - * 2 )- 9/2 (3 + 24*2 + 8*4). Thus, /(0) = 0, /'(0) = 1,
/"(O) = 0, /'"(O) = 1, / <4> (0) = 0, /(5>(0) = 9. Hence, the Maclaurin series is x + $x3 + -jex5 + • • -.
x
39.11 If f(x) = 2 an(x - a)" for \x - a\ < r, prove that In other words, if f(x) has a power
ii -o
series expansion about a, that power series must be the Taylor series for/(*) about a.
f(a) = aa. It can be shown that the power series converges uniformly on | * - a | < p < r , allowing
differentiation term by term:
n(n - 1) • • • [/I - (A: - 1)K(* - «)""*- M we let x = a, fm(a) = k(k-l) 1 • ak = k\ • ak. Hence,
39.12 Find the Maclaurin series for
By Problem 38.34, we know that for |*| <!• Hence, by
Problem 39.11, this must be the Maclaurin series for
39.13 Find the Maclaurin series for tan ' x.
By Problem 38.36, we know that for Hence, by
Problem 39.11, this must be the Maclaurin series for tan *. A direct calculation of the coefficients is tedious.
39.14 Find the Maclaurin series for cosh *.
By Problem 38.43, for all *. Hence, by Problem 39.11, this is the Maclaurin series for
cosh *.
39.15 Obtain the Maclaurin series for cos2 *.
Now, by Problem 38.59, and, therefore, cos 2* =
Since the latter series has constant term 1, and
By Problem 39.11, this is the Maclaurin series for cos2 *.
The benefits of buying summaries with Stuvia:
Guaranteed quality through customer reviews
Stuvia customers have reviewed more than 700,000 summaries. This how you know that you are buying the best documents.
Quick and easy check-out
You can quickly pay through credit card or Stuvia-credit for the summaries. There is no membership needed.
Focus on what matters
Your fellow students write the study notes themselves, which is why the documents are always reliable and up-to-date. This ensures you quickly get to the core!
Frequently asked questions
What do I get when I buy this document?
You get a PDF, available immediately after your purchase. The purchased document is accessible anytime, anywhere and indefinitely through your profile.
Satisfaction guarantee: how does it work?
Our satisfaction guarantee ensures that you always find a study document that suits you well. You fill out a form, and our customer service team takes care of the rest.
Who am I buying these notes from?
Stuvia is a marketplace, so you are not buying this document from us, but from seller jureloqoo. Stuvia facilitates payment to the seller.
Will I be stuck with a subscription?
No, you only buy these notes for $8.63. You're not tied to anything after your purchase.
Can Stuvia be trusted?
4.6 stars on Google & Trustpilot (+1000 reviews)
76449 documents were sold in the last 30 days
Founded in 2010, the go-to place to buy study notes for 14 years now