sheet01.pdf

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course: CalculusI
October5,2006
Problemsheet:Preliminaries
1.Usingthemethodofmathematicalinductionorotherwiseprovethat:
6 ,n2 N ;
(b)1 3 +2 3 +...+n 3 =(1+2+...+n) 2 ,n2 N ;
(c)2 n >n 2 foralln5,n2 N ;
(d)1 2 +3 2 +5 2 +...+(2n+1) 2 = (n+1)(4n 2 +8n+3)
3 ,n2 N ;
(e) 1 1·3 + 1 3·5 + 1 5·7 +...+ 1
(2n−1)(2n+1) = n
2n+1 ,n2 N ;
n+1 ,n2 N ;
(g)sin 3 +sin 2 3 +...+sin n 3 =2sin n 6 sin (n+1)
n(n+1) = n
6 ,n2 N ;
2.Showthat1+ 1 p 2 + 1 p 3 +...+ 1 p n > p nforalln2,n2 N .
3.Supposen2 N ,n2andleta 1 ,a 2 ,a 3 ,...,a n berealnumbersallofthesamesign
suchthata k >−1anda k 6=0fork=1,2,...,n.Showthatthefollowinginequality
holds(1+a 1 )(1+a 2 )·...·(1+a n )>1+a 1 +...+a n .
4.Provethat1− 1 2 + 1 3 − 1 4 +...+ 1
2n−1 − 1 2n = 1
n+1 + 1
n+2 +...+ 1 2n ,foralln2 N .
5.Provethat
(a)1+ 1 p 2 + 1 p 3 +...+ 1 p n >2( p n+1−1),n2,n2 N ;
(b)1+ 1 2 2 + 1 3 2 +...+ 1 n 2 2− 1 n ,n2 N ;
(c) 1
n+2 +...+ 1
3n+1 >1,n2 N ;
6.Showthat3 n >n·2 n forn3.
7.Supposea 1 ,a 2 ,a 3 ,...,a n (n2)representanarithmeticsequence,anda i 6=0for
i=1,...,n.Provethat
1
a n−1 a n = n−1
a 1 a n .
8.Verifythat(a)
n P
n
k
=2 n ;(b)
n P
(−1) k n
k
=0foralln2 N .
k=0
k=0
9.Provethatforx 1 >0,...,x n >0,x 1 +x 2 +...+x n nprovidedx 1 ·x 2 ·...·x n =1
(n2).
n , x 1 0,...,x n 0.
11.Showthatforn2,n2 N (a)n!< n+1
n ;(b)(n!) 2 < (n+1)(2n+1)
6
n
.
2
(a)1 2 +2 2 +...+n 2 = n(n+1)(2n+1)
(f) 1 1·2 + 1 2·3 +...+ 1
n+1 + 1
a 1 a 2 + 1
a 2 a 3 +...+ 1
10.Provetheinequalitybetweenarithmeticandgeometricmeansi.e.
n p x 1 ·x 2 ·...·x n x 1 +x 2 +...+x n
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