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229470536 UNPDF
course: CalculusI
November1,2006
thenR n =a n+1 +a n+2 +...= P 1 k=n+1 a k satisfiesthefollowingestimate:
R n a N+1 q n − N
a n q<1foralln>N.Provethat
1−q whenevernN.
P 1 n=1 a n needstobeaddedinordertoapproximateitssumwiththeaccuracy10 −5 :
(a)
1 P
(n+1)! ,(b)
2 n
1 P
(2n−1)! .
1
n=1
n=1
3.Usingthefactthattheseries P 1 n=1 (−1) n+1
n convergestothesumS(infactS=ln2)
4.Investigatetheconditionalandabsoluteconvergenceoftheseries:
P
(−1) n+1 1 n 2 ,(b)
P
1
(−1) n+1 p n
P
(−1) n
P
1
(−1) n 2n+100
3n+1
n ,
(a)
n+1 ,(c)
n p n , (d)
n=1
n=2
n=1
n=1
(e)
1 P
(−1) n sin 2 n
n ,(f)
1 P
(−1) n p n
n+100 , (g)
1 P
n 2 +(−1) n ,(i)
(−1) n
1 P
(−1) n+1n
n 3 +1 ,
n=1
n=1
n=2
n=1
(j)
1 P
n p+1/n , (k)
1 P
n+(−1) n , (l)
(−1) n
1 P
n 2 p n , (m)
1 P
(−1) n−1 2 n sin 2n a
n .
n=1
n=1
n=1
n=1
5.Estimatethemagnitudeoftheerrorinvolvedinusingthesumofthefirstfourterms
toapproximatetheentiresumoftheseries:(a)
1 P
(−1) n 1 n ,(b)
1 P
(−1) n+11
10 n .
n=1
n=1
6.Findtheseries’radiusanddomainofconvergence.Verifyforwhatvaluesofx2 R
theseriesconvergeonlyconditionally.
P
P
1
(x−2) n
P
(3x−2) n
P
1
(a)
(x+5) n , (b)
10 n , (c)
n (d)
(−1) n (4x+1) n ,
n=0
n=0
n=0
n=0
(e)
1 P
(2x) n (f)
1 P
n+2 , (h)
1 P
n p n3 n ,(i)
x n
1 P
(−1) n (x+2) n
n ,
n=0
n=0
n=0
n=0
(j)
1 P
n! , (k)
1 P
(2x+3) 2n+1
n! , (l)
1 P
p n 2 +3 , (m)
x n
1 P
n p n(2x+5) n
n=0
n=0
n=0
n=1
P
1+ 1 n n x n ,(o)
P
1
(−1) n
n!
e n n x n ,(p)
P
3 n +(−2) n
(n)
n (x+1) n .
n=0
n=1
n=1
Problemsheet:Infiniteseries(cont.)
1.Let P 1 n=1 a n ,a n >0beaseriessuchthat a n+1
2.Usingtheresultfromthepreviousquestionestimatehowmanytermsoftheseries
findthefollowinginfinitesums:
1− 1 2 − 1 4 + 1 3 − 1 6 − 1 8 +...
1
1
(−1) n − 1
(−1) n
1
1
nx n
3 n x n
1
1
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