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Unformatted text preview: Prof. Girardi Math 142 Typical First Few Problems on Sequences & Series Exam If you do not make at least a 50% on this exam’s ﬁrst few problems, then your total exam score will
be only as many points was you managed to get on the ﬁrst few problems. Here are some typical
ﬁrst few problems. 00 1' Fill—in—the blanks / boxes. All series 2 are understood to be 2. =1
Hint: You should NOT write the words absolute no: conditional on Problem 1! 1a. Sequences (Afterall, this is needed for Geometric Series!)
Let —oo < r < oo. (Fill—in—the blanks with exists or does not exist, zle. DNE) ' If M < 1, then Elﬁn—>00 M w—
' If [Tl > 1, then limn_,oo Tn _Dfl_l_3___________
' Ifr=1’thenlimn—>007'n exists (= 1. o If r = 1, then limn_>00 r” DNE ( 05c.) 1b. Geometric Series where —oo < r < 00. The series 2 r”
o converges if and only if r < I o diverges if and only if r Z I la. p—series where 0 < p < 00. The series 2 n—lp o converges if and only if p > I o diverges if and only if p 5 \ 1d. Integral Test for a positive—termed series 2 an where an 2 0.
Let f: [1, 00) —> R be so that oan=f( h. )foreachnEN o f is a EOSH'NC function
. f is a oon+in mo LL$ function . fisa dearcasm or o incrcasu‘n function. Then 2 an converges if and only if 5 :0 10 (X) 4% converges. 1e. Comparison Test for a positive—termed Series Z on where an 2 0.
. HO 3 an 3 bn for all n e Nand gm 00 n Vargas ,then 2a,, con Verges .
oIfOSbnSanforallnENanden chﬂjc‘} ,thenZan Cit/M365 .
1f. Limit Comparison Test for a positivetermed series 2 an where an 2 0.
Let bn > 0 and limn_>00 2—: = . If 0 < L < °O , then 2 an converges if and only if Z L”‘n mvwc’fbé.
1g. Ratio and Root Tests for a positivetermed series 2 an Where an 2 0.
1
Let p = limnnoo a"—+1 or p = limnmoo (an)? , an o If p < J then 2 an converges.
o If p > 1' then Zan diverges.
o If p = 1— then the test is inconclusive. 1h. 1i. 1k. 11. 2 Alternating Series Test for an alternating series Z(—1)"an where an > 0 for each n E N.
If 'an___>__an+1f0reachneN (clecrcmslnj§ o limnnoo an 2 0 then Z(—1)“an M— nth—term test for an arbitrary series 2 an. If limn_,oo an 7é 0_ or limnnoo an does not exist, then 2 an Al Ve—E 3&5 . '. By deﬁnition, for an arbitrary series 2 an, (ﬁll in the blanks with converges or diverges). 0 2 an is absolutely convergent if and only if E anl Com) as
0 2 an is conditionally convergent if and only if 2 an Co Q‘V egg and Z lan ﬁsh/rag
0 2 an is divergent if and only if 2 an clN mag If a power series in :r — 330 has radius of convergence R where 0 < R < 00, then the power series is:
o absolutely convergent for 'X, c R < 96 4 ’X, {a K o divergent for X 4 X9 " R GMA 1 i' R 4 1 Consider a function y = f (ac) where f: [1, 00) —> R . Next consider the corresponding sequence {0,7,}le where an déf' f (n) o If the limit of the function y = f(:r) as so —> 00 is L, then the limit of the correSponding sequence {an}$,°:1 as n ——> 00 is L .
o If limnnoo an 2 L, is it necessarily true that lirnQHOO f (3:) = L? Circle: Yes or Fill in the 3 blank lines (with absolutely convergent, conditional convergent, or divergent) on the
following FLOW CHART for class used to determine if a series 22:17 an is: absolutely convergent, conditional convergent, or divergent. Does 2 [an] converge? ‘fNO a is
12> Z n ngg moi” Since ian Z 0, use a positive term test:
integral test, CT, LCT, ratio/root test. if YES 1; if YES U Is Zen an alternat
ing series? Does 2 an satisfy the conditions of the Al—
ternating Series Test? 2a,, is CmOlli’laanl% Cmgcragat 3. Circle T if the statement is TRUE. Circle F if the statement if FALSE. To be more speciﬁc:
Circle T if the statement is always true and circle F if the statement is NOT always true. Scoring: 2 pts for a correct answer, 1 pt for a blank answer, 0 pts for an incorrect answer. @ F If 2 an converges, then lim,H00 an = 0.
T G?) If limn—aoo an = 0, then 2 an converges. ( L3 ) an = AL)
T @ If a sequence {aﬁfﬁzl satisﬁes that lim,H00 an 2 L and f : [0, 00) —> R is a function satisfying that f (n) = an for each natural number '
then limmnoo f(:c) = L. Con/pedal.” H’X) '—' 3‘“ (THC) . ‘ @ F If a function f: [0, oo) ~—> 1R satisﬁes that limgc_,00 f (cc) = L and
{aﬁﬁozl is a sequence satisfying that f (n) = an for each natural number n, then limnnoo an 2 L. @ F If 2 an converges and 2 bn converge, then Emu + bn) converges.
I
T ® If 201.7, + bn) converges, then 2 an converges and 2 bn converge. 6.2.2 (Ln,1
‘ ‘N 7.17 _ 7.N+1 bu 5F;
Cr) F Ifr¢1andsN=Zrn,thenSN=—1:;—foreachN>17.
n=17 NOTICE, the above sum starts at n = 17, not at n = O. MORE => 4 4. Geometric Series. Let, for N 2 102, N 2n
SN 2 Z 37n+1 '
n=102 4a. Do some algebra to write 3 N as 22,le c 7"” for an appropriate constant c and ratio 7". 4b. Using the method from class (rather than some formula), ﬁnd an expression for s N in closed form
(i.e. without a summation 2 sign nor some dots .). ...
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 Fall '11
 KUSTIN
 Calculus

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