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MAT1322E Test2 VersionA solutions
Typology: Exams
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MAT1322E – Test 2 – Monday, March 4, 2019
Family Name: Student Number:
First Name: †^ Signature:
Version A (^) SOLUTIONS
Circle the most appropriate response regarding this sequence. Show your work!
A. converges to 3 B. diverges to 1 C. converges to ln(3) D. converges to ln(5)
E. converges to e 5 F. diverges to 1 G. converges to e 3 H. none of the previous answers
a 1 =
, a 2 =
, a 3 =
, a 4 =
Find an expression for a (^) n and compute the sum S of the series
n=
an. Show your work!
Circle the most appropriate response:
A. a (^) n = ( 1) n ^1 2 n 3 n^ and S =
B. a (^) n = ( 1) n^2 n ^1 3 n^ and S =
C. a (^) n = 2 n 3 n^ and S =
D. a (^) n = ( 1) n ^1 2 n ^1 3 n^ and S = 1
E. a (^) n = ( 1) n ^1 2 n ^1 3 n^ and S =
F. none of the previous answers
0
lnirnoan
= nltfmaoln( 3M^ - 2) - buff ti (^) )
= figments:^ )
= hinting shift)
for nzl^ , an =f2L
= HEIDI
£7an^
= II If^ }^ )^
" (^) is a geometric series^ with^ first^ term^ a^ = and (^) common ratio^ r =^ - Zz since I^ rt^ = f- If L^1
Fr
= Iz (^) ,
= 1¥ = ÷
's
0
n=
( 1) n^
2 n^ cos(n) 4 n^ n 2
. page 4 of 6
Of the following words, circle all those words (if any) which accurately describe this series.
geometric alternating absolutely conditionally convergent divergent
For each word you circled, you must briefly justify why it describes the series in the space below.
X X^ O (^) x a (^) x
4h -^ h
.^ series^ either^ because^ there^ is^ not^ a^ common^ ratio^ between^ consecutive^ terms
00 Consider (^) Elan I instead^. N (^) =/
o ⇐^ EM "If^ f-
Ey
use Limit^ Comparison^ Test^ on^ this^ series
with bn^ = In
his = figs 4¥.
Since limit (^) exists and (^) equals a (^) positive constant (^) 6=17 (^) , We (^) Know (^) that (^) £ yf÷ behaves^ like^ £7^ In^ by^ virtue^ of the (^) Limit (^) comparison Test^.
£7 In^ is^ a^ geometric^ series^
hence it's^ convergent.^ :^. fi yn¥na^ is^ convergent.
n=
n 3 page 5 of 6
We want to estimate its sum with the sum of its first n terms. How many terms to we need to add so that the error in our estimate R (^) n is guaranteed to be less than 0.01?
A. 7 B. 8 C. 15 D. 14 E. 1 F. 11 G. 10
H. none of the above
Show your work!
Let Fk (^) ) =^ ¥3.^ Then^ f^ is^ positive , continuous^ and^ decreasing for^ all^ x^ >^ o
Rn E^ Sno (^) # DX
Sf # dx^ = find fntxsdx
= this - tax
= figs - ate - tant
= (^) O t I 2h 2
To have (^) Rn L (^) 0.01 it suffices to find n such^ that (^) Sno (^) ¥ d (^) x L^ 0.
2¥ 20.
=) (^) 2¥ L (^) ¥
⇒ (^) 10oz L h
⇒ 50 Ln^ 2
since 72=49 and^ 82=64^ , it^ follows^ that (^) we need n > 8
( since (^) we need n^ >^ Bo^ >^ 549=7 and^ n^ must^ be^ an^ integer)