Solve these limits.
$$\lim_{n\to\infty}\left(\,\frac{1}{n}+\frac{1}{n+1}+\ldots+\frac{1}{2n}\,\right)=?$$
and $\lim_{n \to \infty}a_{n} = ?$ where
$$a_{0}=1\,,\quad a_{n}=\frac{1}{2}\left(\,\frac{2}{a_{n-1}}+a_{n-1}\,\right)$$
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Martin Sleziak
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Hamid Shafie Asl
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3Please consider updating your question with some information about what you have tried or where you are getting stuck. You will find people are much more willing to help if you do! – Nick Peterson Dec 04 '13 at 18:01
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1I want to solve them without using Integral – Hamid Shafie Asl Dec 05 '13 at 05:13
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1Please, post only one question in one post. Posting several questions in the same post is discouraged and such questions may be put on hold, see meta. – Martin Sleziak Mar 24 '14 at 14:10
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1Your second question is answered here: http://math.stackexchange.com/questions/82682/proof-of-convergence-babylonian-method-x-n1-frac12x-n-fracax-n – Martin Sleziak Mar 24 '14 at 14:15
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1Your first question is answered here: http://math.stackexchange.com/questions/73550/is-lim-limits-k-to-infty-sum-limits-n-k12k-frac1n-0 – Martin Sleziak Mar 24 '14 at 14:17
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Thanks. I'll never send more than one question in same post. – Hamid Shafie Asl Mar 24 '14 at 14:25
2 Answers
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Problem 1: compare $$ \lim_{n \to \infty} \sum_{k=n}^{2n}\frac{1}{k} $$ to $$ \lim_{n \to \infty} \int_{n}^{2n}\frac{dx}{x}=\log 2 $$
Problem 2: if the limit exists, it is equal to $$ x=\frac{1}{x} + \frac{x}{2} $$ Can you handle from here?
Alex
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hiint: For the first one you can use the Riemann sum
$$ \frac{1}{n}\sum_{k} \frac{1}{1+k/n} \longrightarrow_{n\to \infty} \int_{0}^{1}\dots dx$$
For the second one, assume $\lim a_n = b$ and subs back in the eq. and solve for $b$.
Mhenni Benghorbal
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Thanks. I can solve them with Integral(answer==$\ln 2$).I want to solve them without Integral.I proved that it is decreasing and bounded below.But I couldn't find it's limit without Integral. – Hamid Shafie Asl Dec 05 '13 at 05:18