Waste less time on Facebook — follow Brilliant.
×

Harmonic series sum

I was wondering on a problem in Alan Tucker's Combinatorics when I noticed this and I wanted to know if it is true(Though I have checked on Wolfram Alpha and it is quite true).

The generating function for \({a}_{r} = \frac{1}{r}\) is of course \(-log(1-x)\).

Therefore, \(h(x) = -log(1-x)\)

Hence, \(h*(x) = \frac{-log(1-x)}{1-x}\) , where \( h * (x) \) is the generating function for the sums of ith coefficient of \( h(x) \).

As a result, the co-efficient of \({x}^{n}\) in \(h*(x)\) is equal to \(1 + \frac{1}{2} + \frac{1}{3}+.... + \frac{1}{n}\).

If this is true, is there any way to find the coefficient of \({x}^{n}\) in \(\frac{-log(1-x)}{1-x}\)?

Note by Kartik Sharma
2 years, 11 months ago

No vote yet
1 vote

  Easy Math Editor

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 \( 2 \times 3 \)
2^{34} \( 2^{34} \)
a_{i-1} \( a_{i-1} \)
\frac{2}{3} \( \frac{2}{3} \)
\sqrt{2} \( \sqrt{2} \)
\sum_{i=1}^3 \( \sum_{i=1}^3 \)
\sin \theta \( \sin \theta \)
\boxed{123} \( \boxed{123} \)

Comments

Sort by:

Top Newest

You mean a closed form? I'm afraid I don't know of one (it probably does not exist). However, the asymptotic limit is \(\gamma+\ln n\), where \(\displaystyle \gamma = \int_{1}^{\infty} \frac{1}{\lfloor x \rfloor}-\frac{1}{x} dx\) is the Euler-Mascheroni constant.

Jake Lai - 2 years, 11 months ago

Log in to reply

Are you sure this is true? *I haven't heard of anything like that. BTW, try finding the coefficient of the \(h(x)\) too(with proof, if you can). Thanks for that info!

Kartik Sharma - 2 years, 11 months ago

Log in to reply

Log in to reply

By \( h * (x) \), do you mean \( h' (x) \)?

Calvin Lin Staff - 2 years, 11 months ago

Log in to reply

No sir, by h*(x), I meant the generating function of the sums of the \({a}_{r}\)s.

Kartik Sharma - 2 years, 11 months ago

Log in to reply

Thanks. I added it in to clarify what that terminology is.

Usually for such Maclaurin expansions, you simply multiply the different parts together.

Calvin Lin Staff - 2 years, 11 months ago

Log in to reply

×

Problem Loading...

Note Loading...

Set Loading...