Page 36 - RMGO 1

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˘
36 Costel BALCAU s , i Mihai Florea DUMITRESCU
s , i, analog,
√
n 2n n ln(2π) + 1 ln(2π)
lim e 2n 12n 2 − 1 = ,
n→∞ e 2e
aplicˆand criteriul cles ,telui obt , inem relat , ia din enunt , .

Aplicat , ii
√
n √ 2n 2n
n! 1 n n n 1
√
√
1) lim = lim n n! − + = .
n→∞ n n→∞ n e e e
√
√ √ √
n n 2n n n 2n n n
2) lim n n! − = lim n n! − + lim −
n→∞ e n→∞ e n→∞ e e
ln n
ln(2π) n ln n ln(2π) e 2n − 1 n ln n
= + lim e 2n − 1 = + lim · ·
2e n→∞ e 2e n→∞ ln n e 2n
2n
ln(2π) ln n
= + lim = +∞.
2e n→∞ 2e
√
p n
3) Pentru s ,irul lui Traian Lalescu L n = n+1 (n + 1)! − n!, n ≥ 2, avem
√
2n+2 √ √
p (n + 1) n + 1 n n 2n n
L n = n+1 (n + 1)! − − n! −
e e
√
2n+2 √
(n + 1) n + 1 n 2n n
+ − ,
e e
deci
√
2n+2 √
ln(2π) ln(2π) (n + 1) n + 1 n 2n n
lim L n = − + lim −
n→∞ 2e 2e n→∞ e e
√
√ 2n+2
n 2n n (n + 1) n + 1
= lim √ − 1
n→∞ e n 2n n
n ln n+1 + ln(n+1) − ln n
= lim e n 2n+2 2n − 1
n→∞ e
n e ln n+1 + ln(n+1) − ln n n + 1 ln(n + 1) ln n
2n − 1
2n+2
n
= lim · ln + −
n→∞ e ln n+1 + ln(n+1) − ln n n 2n + 2 2n
n 2n+2 2n

1 n + 1 ln(n + 1) ln n
= · lim n ln + −
e n→∞ n 2n + 2 2n
n
1 1 n ln(n + 1) − (n + 1) ln n
= · lim ln 1 + +
e n→∞ n 2(n + 1)

1 1 n n + 1 ln n 1
= + · lim ln − = .
e e n→∞ 2(n + 1) n 2(n + 1) e
(n + 1) 2 n 2
4) Pentru s ,irul lui Dumitru M. B˘atinet ,u-Giurgiu b n = p − √ , n ≥ 2,
(n + 1)! n!
n+1 n