Limits and their consequences

I got to this question

limnenk=0nnkk!=12 \lim_{n\rightarrow \infty }e^{-n}\sum_{k=0}^{n}\frac{n^k}{k!} = \frac{1}{2}

I asked some profs, but they handed me solutions using Poisson distribution or lots of integrals. This was supposed to be an exercise by my teacher that could be solved using some basic properties of limits (their arithmetics, squeeze theorem etc.), definition of exe^x as limn(1+xn)n \lim_{n \rightarrow \infty} (1+\frac{x}{n})^n basic limits with e, binomial expansion and logarithms, but without using integrals, series, Stirling formula, asymptotics, Taylor series?

My teacher claims it can be solved with knowledge introduced on lectures so far, which is not much, mainly things mentioned above. Of course, I can use theorems not mentioned on the lectures, but then I have to prove them, and again, with the baisc knowledge. I've been thinking about it for a few days and couldn't do any major progress in my attempts.

Any help would help :-)

Note by Department 8
11 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](https://brilliant.org)example 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}

Comments

Sort by:

Top Newest

Log in to reply

There is some discussion here: https://brilliant.org/problems/wolframalpha-cant-calculate-this-limit/.

Jon Haussmann - 11 months ago

Log in to reply

Well, I tried to look into it but the solution and the available link provided me with nothing of the fact I was looking, as we have not studied that now.

Department 8 - 11 months ago

Log in to reply

×

Problem Loading...

Note Loading...

Set Loading...