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 limn(1n+1+1n+2+...+12n)\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. i=1n1n+i=1n(i=1n11+in).\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 i=1nf(ξi)(xixi1)\displaystyle\sum_{i=1}^n f(\xi_i)(x_i-x_{i-1})? But what if set xi=inx_i=\dfrac{i}{n} and consider the right Riemann sum? Now it will transform into 1n(i=1nf(in))\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 ff we need to consider, so by the definition: limn(1n+1+1n+2+...+12n)=1n(i=1n11+in)=\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)= =1n(i=1nf(in))=011x+1=ln2.=\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. limn(1n+1+1n+2+...+14n)\lim_{n\to\infty}\left(\frac{1}{n+1}+\frac{1}{n+2}+...+\frac{1}{4n}\right)

Problem 3. limn(nn2+1+nn2+4+...+nn2+n2)\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
6 years, 2 months ago

No vote yet
1 vote

  Easy Math Editor

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

  • Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
  • Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
  • Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.
  • Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events.

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold

- bulleted
- list

  • bulleted
  • list

1. numbered
2. list

  1. numbered
  2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1

paragraph 2

paragraph 1

paragraph 2

[example link]( link
> This is a quote
This is a quote
    # I indented these lines
    # 4 spaces, and now they show
    # up as a code block.

    print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.
2 \times 3 2×3 2 \times 3
2^{34} 234 2^{34}
a_{i-1} ai1 a_{i-1}
\frac{2}{3} 23 \frac{2}{3}
\sqrt{2} 2 \sqrt{2}
\sum_{i=1}^3 i=13 \sum_{i=1}^3
\sin \theta sinθ \sin \theta
\boxed{123} 123 \boxed{123}


Sort by:

Top Newest

in problem 3. how we can put the limits from 0 to 3??????????

Attaullah Khan - 6 years, 1 month ago

Log in to reply

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 ln4\ln 4.

Problem 3 can be written as:

limn1n(r=1n11+(rn)2)\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

01dx1+x2=π4\displaystyle \int_{0}^1 \frac{dx}{1+x^2}=\frac{\pi}{4}

Pranav Arora - 6 years, 2 months ago

Log in to reply

Is this from IMC ?

Weijie Chen - 6 years, 2 months ago

Log in to reply

Sorry but I didn't get your question. Can you please explain what is "IMC"?

Pranav Arora - 6 years, 2 months ago

Log in to reply

@Pranav Arora It's International Mathematical Competition and it's the IMO of university studens

Weijie Chen - 6 years, 1 month ago

Log in to reply

bt i dnt undrstand how it is 1 + x^2 & in the problem 1 only x??

Jncy Rana - 6 years, 2 months ago

Log in to reply

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 - 6 years, 2 months ago

Log in to reply

Good job!

Nicolae Sapoval - 6 years, 2 months ago

Log in to reply


Problem Loading...

Note Loading...

Set Loading...