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Unformatted text preview: MATH 138 Assignment 8 g2 pages) Winter 2009 1. *a) *l)) *d) a) *b) (1) Submit ALL problems marked * by 8:20 a.m. Friday, March 20 State the definition of each term: (i) Zen is absolutely convergent.
(ii) 2a,, is conditionally convergent. Justify the statement “One should always check for absolute convergence ﬁrst in considering con
vergence of 2: an." 0° 1 7r
Show that the series Z :5 sm(n§) converges abolutely. n=l 0° cos(mr)
Show that the series Z ——3— converges conditionally. n=l n3
Use the Ratio Test to show that each series either converges absolutely, or diverges oo 00 00 0C!
. (—2)" * _, n+1 nln2 * n" . (—1.1)"
(1) Z n5n+l ('1) Z(_1) (2n)! (“1) E H (1") ET
1t=l n=l 71:1 71:]
0° TL
Show that the Ratio Test fails for the series 2 1 + 2. Then use the Integral Test to determine
‘Il
n=l
whether the series converges or diverges.
00
Use the Integral Test to show that Z 1 converges and Iris sum 5' < I
, . . i "—1 1+ n2 . 2.
00 n ln(n) Show that the Ratio Test fails for the series 2 n3 + 1 . Then prove that the series converges n: 2
. 1
using the Limit Comparison Test With E bn = E —3—.
, n5 3. Use any suitable convergence test, plus general properties of series. to determine whether each series 00
4. a) Estimate Z
n=1 5. *b) *a) converges absolutely, converges conditionally, or diverges. 0° lnn 0° +1 7122" 0° 61/" 0° 2 e"
*3) EH)" 7 b) X (—1)" T C) E a *d) X (a? + one)
11:2 11:1 ' n=l n=l '—
00 . DO
2+smn 1+(—1)"
*e) 2a,, where (11 = 1, an+1 = Tan f) "2:1 —ng—
n: ‘ = with error less than 0.001. (_1)n+1
n3 Determine the least number of terms of the series in problem 321) (above) needed to estimate the
sum with error less than 0.01. (Note how slowly this series converges.) °° bn Show that —' converges absolutely for all I) E R. and explain why this implies
n. n=l
b"
lim 5 = 0 for all b E R (i.e.. n! grows faster than any exponential, eventually).
71—900 " , 5n b) What is the cut0H value K which guarantees —— < 1 for n > K? n! MATH 138  Winter 2009 Assignment #8 Page 2 of ‘2 6. Find the interval of convergence (including a check of the endpoints) for each of the given power series: 00 00 oo 00
M23; — 1)” 2nx" 1 nlx"
a) 2—3" W Z n. *C) 237(“0” d) E: 2,.
11:1 11:0 . n.:=l 71:]
m 00 v 00 oo .
* . _ cos(n7r)m” 3nx" (2x — 1)"
e) §_jn(n+1)(z 1)" f) EjT g) 2 n2 *h) 2— g
11—] "=0 11:1 11:] 7. Use GST to write each of the given functions as a power series centred at :1: : a. and state the interval
of convergence. a) f(m)=2x1+3’a=0 b) f($)=:r:2’a=—1
*c) f(a:)=2_:$.a=2 *d) f(:c)=1+1813,a=0 8. For each of the following, either prove the statement is TRUE or give a counterexample showing it is co
FALSE. [HINTI All of these can be done using selected examples of the pseries 2 % and/or the 71:1 1)n—] 00
alternating pseries Z ( np 11:] a) If lim lanl = 0, then Zan converges.
00 11—. *b) if 2a” is a divergent series, then REMINDER
liin an aé 0. Term Test 2 on March 23 will
"H00 cover problems 611 of Assignment 5
c) If 2 an converges, then 2 a3, converges. plus all of Assignments 6 and 7, and
* . an“ Assignment 8 except for problems 6
d) If hm = 1, then Zen converges. and 7. n—wo a." *e) If lim n2 (in = 1, then 2 a." converges.
Tl—’OO CHALLENGE: 9. a) Two players take turns ﬂipping a coin. The game is won by whomever ﬂips the ﬁrst head. Show
that, if the coin is fair. then the probability that the player who gets the ﬁrst turn wins the game
is l l 1
2 1
s + 32 + ”'+ 2211—1 p: + +~ b) What is the value of p? Does this surprise you? W hy/not? ...
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This note was uploaded on 09/04/2009 for the course MATH 138 taught by Professor Anoymous during the Winter '07 term at Waterloo.
 Winter '07
 Anoymous
 Calculus

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