Summation Series to Product Series Part 2.

In the first part of these notes we showed the following S(n)=k=1k=nf(k)k!=k=2k=n(1+f(k)ak)=P(n) S(n) = \sum_{k = 1}^{k = n}\frac{f(k)}{k!} = \prod_{k = 2}^{k = n} (1 + \frac{f(k)}{a_k}) = P(n) for all n>1n > 1 where the sequence (ak)(a_k) is given by the recurrence relation a2=2 a_2 = 2 and ak+1=(k+1)(ak+f(k))a_{k+1} = (k+1)(a_k + f(k)).

In this note we'll have a look at some applications of this.

Part 1 : A couple of Examples

For this constant function we have S(n)=k=1k=n1k!S(n) = \sum_{k = 1}^{k = n}\frac{1}{k!} . So k=2k(1+1ak)=e1 \prod_{k = 2}^{k \rightarrow \infty} (1 + \frac{1}{a_k}) = e-1 where a2=2 a_2 = 2 and ak+1=(k+1)(ak+1)a_{k+1} = (k+1)(a_k +1) by considering Taylor expansion of ee.

Here is another example : we can write the harmonic series as n=1n1n\sum_{n=1}^{n \rightarrow \infty} \frac{1}{n} . So k=2k(1+(k1)!ak) \prod_{k = 2}^{k \rightarrow \infty} (1 + \frac{(k-1)!}{a_k}) where a2=2 a_2 = 2 and ak+1=(k+1)(ak+(k1)!)a_{k+1} = (k+1)(a_k +(k-1)!)

Part 2 : Questions and Remarks for Brilliant Here's some questions I have for the Brilliant Community.

1) We showed that we can re-write the harmonic series as k=2k(1+(k1)!ak) \prod_{k = 2}^{k \rightarrow \infty} (1 + \frac{(k-1)!}{a_k}) . Can we prove that this diverges without assuming the summation series n=1n1n\sum_{n=1}^{n \rightarrow \infty} \frac{1}{n} ?

2) Can we re-write the sequence a2=2a_2=2 and ak+1=(k+1)(ak+f(k))a_{k+1} = (k+1)(a_k +f(k)) with out recursion? Do these recursive sequences have any combinotorical properties?

3) We haven't really learnt anything significantly new from re-writing these series and in product series form this is computationally difficult due to the recursive sequences.

4) Post any interesting examples you can re-write below!

Note by Roberto Nicolaides
6 years, 5 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}


There are no comments in this discussion.


Problem Loading...

Note Loading...

Set Loading...