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Playing with Integrals: Limits and Integrals 1

As I said in previous post the formal definition of the Riemann Integral is very useful when solving Olympiad problems.

Problem 1. Find the following limit \[\lim_{n\to\infty}\left(\frac{1}{n+1}+\frac{1}{n+2}+...+\frac{1}{2n}\right)\]

Solution. Let try to make our sum into something more or less similar to Riemann sum. \[\sum_{i=1}^n\frac{1}{n+i}=\frac{1}{n}\left(\sum_{i=1}^n\frac{1}{1+\frac{i}{n}}\right).\]

Does this remind you of the monstrous \(\displaystyle\sum_{i=1}^n f(\xi_i)(x_i-x_{i-1})\)? But what if set \(x_i=\dfrac{i}{n}\) and consider the right Riemann sum? Now it will transform into \(\dfrac{1}{n}\left(\displaystyle\sum_{i=1}^n f\left(\dfrac{i}{n}\right)\right)\).

From the last formula we can easily understand what function \(f\) we need to consider, so by the definition: \[\lim_{n\to\infty}\left(\frac{1}{n+1}+\frac{1}{n+2}+...+\frac{1}{2n}\right)=\frac{1}{n}\left(\sum_{i=1}^n\frac{1}{1+\frac{i}{n}}\right)=\] \[=\dfrac{1}{n}\left(\displaystyle\sum_{i=1}^n f\left(\dfrac{i}{n}\right)\right)=\int^1_0\frac{1}{x+1}=\boxed{\ln 2}.\]

Now using the same approach try solving the following problems.

Problem 2. \[\lim_{n\to\infty}\left(\frac{1}{n+1}+\frac{1}{n+2}+...+\frac{1}{4n}\right)\]

Problem 3. \[\lim_{n\to\infty}\left(\frac{n}{n^2+1}+\frac{n}{n^2+4}+...+\frac{n}{n^2+n^2}\right)\]

Note by Nicolae Sapoval
3 years, 1 month ago

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Problem 2 is same as Problem 1, the only difference is that the integration limits change. The lower limit is 0 and upper limit is 3. Hence, the answer is \(\ln 4\).

Problem 3 can be written as:

\(\displaystyle \lim_{n\rightarrow \infty} \frac{1}{n}\left(\sum_{r=1}^n \cfrac{1}{1+\left(\frac{r}{n}\right)^2} \right) \)

The above is equivalent to

\(\displaystyle \int_{0}^1 \frac{dx}{1+x^2}=\frac{\pi}{4} \) Pranav Arora · 3 years, 1 month ago

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@Pranav Arora bt i dnt undrstand how it is 1 + x^2 & in the problem 1 only x?? Jncy Rana · 3 years, 1 month ago

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@Jncy Rana Because to get the answer we have to consider a variable (say x) Which will be in form of r/n after modifying the problem. Manish Mishra · 3 years, 1 month ago

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@Pranav Arora Good job! Nicolae Sapoval · 3 years, 1 month ago

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@Pranav Arora Is this from IMC ? Weijie Chen · 3 years ago

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@Weijie Chen Sorry but I didn't get your question. Can you please explain what is "IMC"? Pranav Arora · 3 years ago

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@Pranav Arora It's International Mathematical Competition and it's the IMO of university studens Weijie Chen · 3 years ago

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in problem 3. how we can put the limits from 0 to 3?????????? Attaullah Khan · 3 years ago

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