37 lines
1.2 KiB
TeX
37 lines
1.2 KiB
TeX
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\begin{deloppgaver}
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\delo
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\begin{align*}
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a_n &= \frac{n}{n+1} \\[1ex]
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a_1 &= \frac{1}{1+1} = \frac{1}{2} \\[1ex]
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a_2 &= \frac{2}{2+1} = \frac{2}{3} \\[1ex]
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a_3 &= \frac{3}{3+1} = \frac{3}{4} \\[1ex]
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a_4 &= \frac{4}{4+1} = \frac{4}{5} \\[1ex]
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a_{100} &= \frac{100}{100+1} = \frac{100}{101} \\[1ex]
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\end{align*}
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Jeg gjetter at \[\lim_{n \to \infty} a_n = 1\]
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\delo
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\begin{align*}
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a_n &= \frac{n^2}{n+1} \\[1ex]
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a_1 &= \frac{1^2}{1+1} = \frac{1}{2} \\[1ex]
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a_2 &= \frac{2^2}{2+1} = \frac{4}{3} \\[1ex]
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a_3 &= \frac{3^2}{3+1} = \frac{9}{4} \\[1ex]
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a_4 &= \frac{4^2}{4+1} = \frac{16}{5} \\[1ex]
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a_{100} &= \frac{100^2}{100+1} = \frac{10000}{101} \\[1ex]
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\end{align*}
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Jeg gjetter at \[\lim_{n \to \infty} a_n = \infty\]
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\delo
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\begin{align*}
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a_n &= \frac{n}{n^2+1} \\[1ex]
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a_1 &= \frac{1}{1^2+1} = \frac{1}{2} \\[1ex]
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a_2 &= \frac{2}{2^2+1} = \frac{2}{5} \\[1ex]
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a_3 &= \frac{3}{3^2+1} = \frac{3}{10} \\[1ex]
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a_4 &= \frac{4}{4^2+1} = \frac{4}{17} \\[1ex]
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a_{100} &= \frac{100}{100^2+1} = \frac{100}{10001} \\[1ex]
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\end{align*}
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Jeg gjetter at \[\lim_{n \to \infty} a_n = 0\]
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\end{deloppgaver}
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