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Unformatted text preview: MATH 138 Assignment 9 (2 pages) Winter 2007 Submit all problems marked * on Friday, March 30 1. Find the interval of convergence (including a, check of the endpoints) for each of the given power series: 00 DC‘ DC 00
71(21: — 1)" 2711'” * 1 mix” a) 2—3“ W Z n! C) Z3 (7" +1)” d) 2 2” 11:1 n:0 n=l n21 CO 00 oo 00 cos(mr):r” 371:3" (23: — 1)” * ‘ _ TL f *1 __
e) "i=1n(n+ 1)(:r 1) ) ”:0 4" g) n; n2 1) ”5:1 ﬂ 2. Use GST to write each of the given functions as a power series centred at a: = a, and state the interval
of convergence, 1 3
* 1 3F _ 1 _
c) f(l)—2+m.a—2 d) f<m>~1i+8$3,a—o
3. *a.) Use a known series to ﬁnd the Maclaurin series for f(3:) = 1: 2. Hence, by inspection, find
.1
f‘9)(0) .2 b) Repeat part a) for f(:z) = 3226—“ . 4. Use known Maclaurin series to ﬁnd a Macleurin series for each of the given functions. State the radius of
convergence (from the theorems used), and then check the endpoints to ﬁnd the interval of comrergence. *a‘) f(x) = {136—333 *b) f(a:) =1n(2 + 222) *c) f(m) = /$ sin(t3)dt d) f(.r) = arctan :1:
0 .'L‘ $2
e>f<x)=ln(1+) webﬁfa *g)f(w)=:vCOS(2> mm): 1 1—3: *i) erf (3:) = ifze—ﬁdt j) f(r) = /m 1 dt k) f(3:) = ﬂ *1) f($) : /2 arctan(t2)dt
ﬁe ‘ 04+t3 6—:r—zr2 .0 ,
HINTS:
‘2
b) 1n(2 +1.2) : ln (2 (1 + 73)) and use Example 11 on page 107 of Course Notes; m 1
d) arctanx=/ , dt.
0 1+1?
u e) In (—) = lnu — 1n 1), ’U 1 1—23' f) Differentiate the series for
k) Use a partial fractions decomposition. 5. *a) Use a Maclaurin series to evaluate sin(1) with error less than 2 x 10—4. 1 00 00
*b) If 1+3: = Z(—1)”m", what function is represented by Z(1)"+ln:r"_l, and on what
71:0 n:1
interval? . 7T.— . . . .
c) Does secs: have a Taylor series about a. = 5.” Why? Find the ﬁrst three non—zero terms of the Maclaurin series for sec m. 10 (1) Use known Maclaurin series to prove Euler’s formula: (2 = c056 + 21 sin 6, where 12 = ~1. MATH 138 — Winter 2007 Assignment #9 Page 2 of 2 2
6. Show that f (1,) =Zn332”_ .Hence sum the series —T—L. _ 5x2)2 4n
n=1 1 HINT: Start by ﬁnding the series for 1 $2 . Then use a suitable operation to get the series for f(:r). blll L
. . . . . . . . for :r 0
*7. A function used in digItal Signal processmg 1s s1nc(a:) = (1; # . It does not have an
1 for :L' = 0
antiderivative expressible in terms of elementary functions. a) Show that, if you divide the known Maclaurin series for sinx by x, the resulting series has value
1 when :r = 0, and hence is the Maclaurin series for sine 1‘.
1 b) Evaluate the integral I = / sinc (aching with error less than 0.001.
. 0 00 .
. . 2w — 1 "'
8. Given the series 2%—
71:1
radius of convergence R : 1, and determine the interval of convergence for such I). , where b > 0 is constant, ﬁnd the value(s) of I) such that the series has *9. a) Show that the circumference of the ellipse
x = (asinB, bcosﬁ) for 0 S 0 S 27r, where a > b > 0, is given by _ 2
— Lila/W2 V 1 — e2 sin 2,0(10 where e— — ————b. CL 2 b) Use the binomial expansion for (1 + 9:)1/2, with :r = —e sin2 0, to write the ﬁrst four terms of the Maclaurin series for the integrand. '7r/2 . .I'... r _
,0 246~2'ri 2 c) Use the formula and the approximation from b) to evaluate L up to terms in (26. For more practice, see text pages 753 (#1 — 28), 759 (# 1  32, 38, 39), 770 (# 11 — 18, 23 — 30, 39 — 46) ...
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