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Lecture 21

MAT187H1 Lecture 21: 9.6 Alternating Series and Absolute and Conditional Convergence 10.1 Approximating Functions with PolynomialsPremium


Department
Mathematics
Course Code
MAT136H1
Professor
Anthony Lam
Lecture
21

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MAT187H1Lecture219.6AlternatingSeriesandAbsoluteandConditional
Convergence,10.1ApproximatingFunctionswithPolynomials
Testforconvergence: ..
k=0 k!
(−1)k= 1 − 1 + 1
2! 1
3! +1
4! − .
Since(k+1)!>k! (ie.termsaredecreasing),and
1
k!>1
(k+1)! lim
k→∞
1
k!= lim
k→∞
1
k(k−1)(k−2)... = 0
seriesconvergesbyAlternatingSeriesTest
Note:(inchapter10) e
1
k=0
1
k!=e
1
=1/e
k=0 k!
(−1)k=
ApproximatingSumsofConvergentAlternatingSeries
ConsiderconvergentA.S.
a
1
a
2
+a
3
a
4
+a
5
...(− ) a
k=0 1k+1
k=
DefineRemainderR
n
SS
n
OddtermsofS
n
are>SieSS
odd
<0
EventermsofS
n
are<SieSS
even
>0
TheoremForconvergentseries (whichconvergestoS),then
th
termremainder,(− ) a
k=1 1k+1
k
R
n
=SS
n
hasthesamesignas(n+1)
th
termand0≤|R
n
|<a
n+1
ExampleGiven (− ) ..
k=0 1k1
2k+1 = 1 − 3
1+5
17
1+ . = 4
π
Howmanytermsareneededtoestimate toaccuracyof10
4
?(iefor|R
n
|=10
4
,n=?)
4
π
Foranyn,|R
n
|<a
n+1
=1
2(n+1)+1 =1
2n+3
for <10
4
<>2n+3>10
4
1
2n+3
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