Download math 129 exam 1 document solutions and more Exams Mathematics in PDF only on Docsity!
Math 129 Exam 3 Fall 2024 Name (Please Print.): You need to show all your work to get a full credit. Problem 1 (5 points each) Determine whether the series converges or diverges. Give a reason such as a test.
(a) n 1
cos 2 n n^2 3 n
(b) n 1
(^2) n^2 3 n^3 4 n 1
(c) n 1
3 n n 5 n Diverge general term (^) :Eo (^) [+ 3 n) n →^ ∞ (^) , 3 m= -^3 - → o since (^) cos (x)oscillates betuwen^ -^ taud^1 ,the senies dost (^) o fythe necessaryconditionbrcon^ eg is^ a^ L an^ #^0 ) (^) , thereforer,^ it d's irerges
highest - degree ionthenaoom^ &darom, 24
~ (^) = ,^ which^ diverges.
By the^ limitComparionTest^ ,thegivemseriesis
divege onverges . Bominant Jeron (^). .~ (^) ( ) This is aagronatra^ series^ wite^ γ=^5 L^1.^ which^ Eonwerges. By the^ Limit^ ComparrtsTestwith^ ) " ,
the given seises^ _conwerages
Problem 2 (5 points) Determine whether the series n 1
1 n ^1 2 n 2 n^4 3 n^3 1 converges
absolutely. Also find the maximum possible error if n 1
1 n ^1 2 n 2 n^4 3 n^3 1 is approximated by S 4.
Problem 3 (5 points) Find the sum of the series n 1
5 ^3 n 4 n ^1 , if it converges. 2 Determine whether^ the^ series^ converges absolutely : an 2 Ʃ )^ n (^4) + (^3) n 3 tt Ʃ (^) converges
because it is a
Dominant P
×
. "□= (^2) as k (^) ts anmjsngm a Termms FInd (^) maximum possible emor if^ S^4 is^ used^ : Eror ≤^ lak+^11 , an =*^3.
s 4 , a ; =^ ≈(^55
) 4
- 3 ( 5 )+)= → (^25 ). 25 +^37 s^1 =^. 0 ≈ (^0). (^04995). 6
.^ themaximumpossible^ eronwhenusig^ Se^ is^ approximately 0. 04 5 ." π (^3)
= .( ) ( ) ) For (^) a geometric
eies to^
convenge ,^ the^ absolute value^ of^ r^ must fit wth^ r^1 4. =
. Thisseq es c =.
,
, , ( (^) ) (^) , × (^). μ m = . C
Problem 6 (6 points) Find the Taylor series of f x sin x in x . Write down the four terms of the Taylor series. Do it by direct computation. Problem 7 (a) (3 points) Using the Taylor series of ex^ in x , find the Taylor series of xe x 2 in x.
(b) (4 points) Approximate
0 1 e x 2 dx using the first four nonzero terms of a Taylor series of e x 2 in x.
a =^ π^ f(a)^ =^ sin^ a)^ f(x)^ =^ f^ (π)^ ←^ t^
' (xx- ) +
- " : (x-x) p … f(x)=^ simx (^) , f(a) =^ sin^ (a)^ =^0 +(←^ ) =^ O
f'^ (c)=^ Cosx^ ,f^ )^ (x )=[^ os(x^ )=-^1
f '^ (a)^ =^ -^1 f" (x) =^ - sin, (^) f(π]^ =^
. (^) sin (π) =^ o f " (π (^) ) -^0 f (^) (x) (β^3 =
- sosc] , fi"[a )^ ÷ - cos)
f "' ( )a f( 4 ) c)^ : sin (^) x,^ f^ @" (π) = (^) sin (e) = (^) o ∴ f(s)=^0 -^ (x)+ .(x-π^ )^ (x " repeat (^) cylically
fin'(^ x) = sinx,^ cosx^ ,-si^ cosc)
fG) =-c-λ+← . = xe "^ ' = ②. . " = . " e
- (^) π^ ' = . " " 1 - x+ ③ : - :
S. ee
'bs= f :[ 1 -^ π
[ ] =.
5 : 1. da) -
S!^ π^ '+ /: (^) : '
[ ]!=
. s = (^) = =
Problem 8 (a) (5 points) Find the power series expansion of x 1 3 x in x , and also find the radius of convergence. (b) (6 points) Find the power series expansion of 1 1 2 x and find the power series expansion of ln 1 2 x in x using term by term integration of power series expansion of 1 1 2 x
- , lef γ=^3 x (^) ,
- (^) .( 3 a)" =^. 3 mxu^ , (^13) l 4 or bal<
- =- ".^ 3 nx μ (^) = ) 3 m + x (^) ) m^ =^ ntl
- =
)^3 m^ "
x ". O ① = =.
) " 2 μ x " ,^ 2 , = . ② |^ n^ (^1 +^2 )^ = S *u = S . ]uznsaud. = S
]
" (^2) " x "d =. S ) znxndsc =
. " C
