kolo.docx rozw .pdf
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Obliczy¢:
12 +32 +...+(2n−1)2
.
n3
n→∞
lim
Wskazówka:
12 + 22 + 32 + ... + n2 =
n(n+1)(2n+1)
.
6
Rozwi¡zanie:
Sum¦ szeregu kwadratów liczb nieparzystych mo»na przedstawi¢ jako ró»nic¦ sum szeregu kwadratów liczb
naturalnych oraz ró»nic¦ sum szeregu kwadratów liczb parzystych.
2n
∑
Suma szeregu kwadratów liczb naturalnych (zwó¢ uwag¦ na 2n zamiast n!):
n
∑
Suma szeregu kwadratów liczb parzystych:
(2k)2 =
k=1
4·
n
∑
(22 · k 2 ) = 22 ·
k=1
k2 =
2n(2n+1)(4n+1)
6
(z wskazówki)
k=1
n
∑
k 2 = 4 · (12 + 22 + 32 + ... + n2 ) =
k=1
n(n+1)(2n+1)
.
6
Suma szeregu kwadratów liczb nieparzystych:
12 + 32 + ... + (2n − 1)2 =
2n
∑
k2 −
k=1
Wracamy do granicy:
12 +32 +...+(2n−1)2
n3
n→∞
lim
= lim
n→∞
n
∑
(2k)2 =
2n(2n+1)(4n+1)
6
−4·
n(n+1)(2n+1)
6
=
n(2n+1)(2n−1)
3
k=1
n(2n + 1)(2n − 1)
3n3
(4n3 −n)
3
n→∞ 3n
= lim
1
n3 ·(4− 12 )
n
3n3
n→∞
= lim
4− 12
n
n→∞ 3
= lim
=
4
3
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