oblicz granice

[katharsis223]

oblicz granice ciągu o wyrazie ogólnym:

a)\(\displaystyle{ a_{n} = \sqrt[3]{ \frac{1-27n^{2} }{8n^{2}-1 } }}\)

b)\(\displaystyle{ b_{n} = (\sqrt{5n+3}-\sqrt{5n})}\)

c)\(\displaystyle{ c_{n} = (\sqrt{n^{2}+2n-3}-n)}\)

d)\(\displaystyle{ d_{n} = \sqrt[n]{2^{n}-3*4^{n+5*7^{n} } }}\)

e)\(\displaystyle{ e_{n} = \sqrt{3^{n}+5*6^{n} }}\)

f)\(\displaystyle{ f_{n} = \sqrt[n]{ (\frac{1}{3})^{n} + (\frac{1}{4})^{n} + (\frac{1}{5})^{n} }}\)

g)\(\displaystyle{ g_{n} = \sqrt[n]{ (\frac{2}{9})^{n} + (\frac{4}{9})^{n} + (\frac{7}{9})^{n} }}\)

h)\(\displaystyle{ h_{n} = (\frac{1-n^{2} }{5-n^{2} }) ^{2n}}\)

i)\(\displaystyle{ i_{n} = (\frac{3n+2}{4n+3}) ^{2n}}\)

[raphel]

c)\(\displaystyle{ ...= \sqrt{n ^{2} +2n -3 } -n \frac{\sqrt{n ^{2} +2n -3 } +n}{\sqrt{n ^{2} +2n -3 } +n} = \frac{n ^{2} +2n-3 -n ^{2} }{\sqrt{n ^{2} +2n -3 } +n} = \frac{2n-3}{n \sqrt{1+ \frac{2}{n} - \frac{3}{n ^{2} } }+n } 2}\)

b) podobnie jak c
f) , g) z trzech ciągów
\(\displaystyle{ \sqrt[n]{ (\frac{1}{5}) ^{n} } qslant \sqrt[n]{ (\frac{1}{3}) ^{n} +(\frac{1}{4}) ^{n} +(\frac{1}{5}) ^{n} } qslant \sqrt[n]{3 (\frac{1}{5}) ^{n}} \frac{1}{5}}\)

[beatka-k16]

\(\displaystyle{ i_{n} = ( \frac{3n+2}{4n+3}) ^{2n}= \left( \frac{4n+3-n-1}{4n+3}\right)^{2n}= \left( 1+ \frac{(-n-1)}{4n+3} \right)^{2n} = \left[ \left( 1+ \frac{(-n-1)}{4n+3} \right)^{ \frac{4n+3}{-n-1}} \right]^{ \frac{(-n-1)}{4n+3} \cdot 2n}= \left[ \left( 1+ \frac{(-n-1)}{4n+3} \right)^{ \frac{4n+3}{-n-1}} \right]^{ \frac{-2n^2-2n}{4n+3}}= e^ {-\frac{1}{2}}\)