Oglejmo si naslednje verižne ulomke
![\[x_1=1+\cfrac{3}{2+\cfrac{3}{2+\cfrac{3}{2+\cfrac{3}{\ddots}}}},\]](http://vincenc.petruna.com/wp-content/ql-cache/quicklatex.com-9275a4657a550f32d244591411f5b526_l3.png "Rendered by QuickLaTeX.com")
![\[x_2=1+\cfrac{8}{2+\cfrac{8}{2+\cfrac{8}{2+\cfrac{8}{\ddots}}}},\]](http://vincenc.petruna.com/wp-content/ql-cache/quicklatex.com-af9bee972c212b1c13e84887caa7c7f2_l3.png "Rendered by QuickLaTeX.com")
![\[x_3=1+\cfrac{15}{2+\cfrac{15}{2+\cfrac{15}{2+\cfrac{15}{\ddots}}}},\]](http://vincenc.petruna.com/wp-content/ql-cache/quicklatex.com-ccc34a39a8b43b3bcb144f948b8f3861_l3.png "Rendered by QuickLaTeX.com")
![\[x_4=1+\cfrac{24}{2+\cfrac{24}{2+\cfrac{24}{2+\cfrac{24}{\ddots}}}},\]](http://vincenc.petruna.com/wp-content/ql-cache/quicklatex.com-1f12872b3b40cdb858315285a48b55b5_l3.png "Rendered by QuickLaTeX.com")
![\[x_5=1+\cfrac{35}{2+\cfrac{35}{2+\cfrac{35}{2+\cfrac{35}{\ddots}}}},\]](http://vincenc.petruna.com/wp-content/ql-cache/quicklatex.com-507777b1b670b99608abb4782b98bd5b_l3.png "Rendered by QuickLaTeX.com")
itd.
Najprej opazimo, da se ti verižni ulomki razlikujejo samo v števcih, zato najprej pomislimo, da so morda narejeni na enak način, po isti šabloni. Vrednost ulomkov lahko tudi najprej uganemo, npr, tako, da izračunamo nekaj zaporednih približkov vsakega od njih. Dobimo naslednje vrednosti
![\[x_i=i+1;\qquad i=1,..,5\]](http://vincenc.petruna.com/wp-content/ql-cache/quicklatex.com-8f5f7eb779e35fa840ba2bfbd5447898_l3.png "Rendered by QuickLaTeX.com")
Nato si ogledamo te vrednosti ter števce verižnih ulomkov. Dovolj oster pogled razkrije, da so števci v verižnem ulomku 
enaki 
torej
![\[x=1+\cfrac{x^2-1}{2+\cfrac{x^2-1}{2+\cfrac{x^2-1}{2+\cfrac{x^2-1}{\ddots}}}}\]](http://vincenc.petruna.com/wp-content/ql-cache/quicklatex.com-e31a1d805f7794393c242a03fc8d0dee_l3.png "Rendered by QuickLaTeX.com")
ali
![\[x=1+\cfrac{x^2-1}{1+1+\cfrac{x^2-1}{1+1+\cfrac{x^2-1}{1+1+\cfrac{x^2-1}{\ddots}}}}.\]](http://vincenc.petruna.com/wp-content/ql-cache/quicklatex.com-a29215b3105f370a961c912df9f7606d_l3.png "Rendered by QuickLaTeX.com")
Od tod pa hitro prepoznamo
![\[x=1+\frac{x^2-1}{1+x},\]](http://vincenc.petruna.com/wp-content/ql-cache/quicklatex.com-4ccb4d0360f9b05b6169caf2616d6837_l3.png "Rendered by QuickLaTeX.com")
oziroma znan obrazec iz osnovne šole
![\[x-1=\frac{x^2-1}{1+x}.\]](http://vincenc.petruna.com/wp-content/ql-cache/quicklatex.com-523a3aff360c0c85cef77851199a30fb_l3.png "Rendered by QuickLaTeX.com")
To je tisto kopito, po katerem so sestavljeni zgornji verižni ulomki. Sestavite še kakšnega sami.
Naloga: Tudi verižni ulomek
![\[[1;1,\overline{2}]=1+\cfrac{1}{2+\cfrac{1}{2+\cfrac{1}{2+\cfrac{1}{\ddots}}}}\]](http://vincenc.petruna.com/wp-content/ql-cache/quicklatex.com-ff95c3f5cf911da1f914e1a6701ceb7a_l3.png "Rendered by QuickLaTeX.com")
spada v to skupino. Zakaj? Kolikšna je njegova vrednost? Sestavi še kakšen tovrstni verižni ulomek iracionalne vrednosti.
![\[(a^2-b^2)(c^2-d^2)=(ac+bd)^2-(ad+bc)^2,\]](http://vincenc.petruna.com/wp-content/ql-cache/quicklatex.com-a429338353be442cf0c0da40eeace9ca_l3.png "Rendered by QuickLaTeX.com")
![\[(a^2-Nb^2)(c^2-Nd^2)=(ac+Nbd)^2-N(ad+bc)^2,\]](http://vincenc.petruna.com/wp-content/ql-cache/quicklatex.com-a6f712b9c77eb11f5b057827f983d996_l3.png "Rendered by QuickLaTeX.com")
![\[(1+2+3+\dots +n)^2=1^3+2^3+3^3+\dots +n^3,\]](http://vincenc.petruna.com/wp-content/ql-cache/quicklatex.com-bfb6507b85355a228fbaf36b3e911c78_l3.png "Rendered by QuickLaTeX.com")
![\[{\left(1+\frac{1}{3}-\frac{1}{5}-\frac{1}{7}+\frac{1}{9}+\frac{1}{11}-\dots\right)}^2=1+\frac{1}{3^2}+\frac{1}{5^2}+\frac{1}{7^2}+\dots,\]](http://vincenc.petruna.com/wp-content/ql-cache/quicklatex.com-bf6ac377be6bb9b0734a8948e989c334_l3.png "Rendered by QuickLaTeX.com")