Another divergence proof

Here is another proof for the divergence for the harmonic series . I assume no one has found it earlier! (its awfully dumb to say I'm the first one!!)..... consider the integral I(a,x)=∫0x(sin⁡x)acos⁡xdx I(a,x) = \int_0^{x} \frac{(\sin x)^{a}}{\cos x} dx we will tease this until it cries out an interesting series .....

multiplying 1 continuously OR (sin⁡x)2+(cos⁡x)2 (\sin x)^2 + (\cos x)^2 yields =∫0x((sin⁡x)acos⁡x)((sin⁡x)2+(cos⁡x)2)......dx = \int_0^{x} (\frac{(\sin x)^{a}}{\cos x})((\sin x)^2 + (\cos x)^2) ...... dx =∫0x(sin⁡x)acos⁡x+((sin⁡x)a+2cos⁡x)((sin⁡x)2+(cos⁡x)2)dx = \int_0^{x} (\sin x)^{a}\cos x + (\frac{(\sin x)^{a+2}}{\cos x})((\sin x)^2 + (\cos x)^2) dx =∫0x((sin⁡x)acos⁡x+(sin⁡x)a+2cos⁡x+(sin⁡x)a+4cos⁡x.......∞)dx = \int_0^{x} ( (\sin x)^{a}\cos x + (\sin x)^{a+2}\cos x + (\sin x)^{a+4}\cos x ....... \infty ) dx =((sin⁡x)a+1a+1+(sin⁡x)a+3a+3+(sin⁡x)a+5a+5.....∞) = ( \frac{(\sin x)^{a+1}}{a+1} + \frac{(\sin x)^{a+3}}{a+3} + \frac{(\sin x)^{a+5}}{a+5} ..... \infty ) plug in a=1;x=π2 a=1 ; x = \frac{\pi}{2} =1a+1+1a+3+1a+5......∞ = \frac{1}{a+1} + \frac{1}{a+3} + \frac{1}{a+5} ...... \infty to get =12(11+12+13......∞) = \frac{1}{2} ( \frac{1}{1} + \frac{1}{2} + \frac{1}{3} ...... \infty ) which is half the harmonic series ........ solving the integral for the same a=1 gives I(1,π2)=∫0π2tan⁡xdx I(1,\frac{\pi}{2}) = \int_0^{\frac{\pi}{2}} \tan x dx =∞ = \infty ....... hence the harmonic series diverges!

What is more important here is the behavior of the series as the limit transforms from 0 to x ..... When x starts from 0 further (keeping a as 1 ) and reaches pi/2 the series changes its values according to the function tan x ....... beautiful! isn't it ?

please post your comments and ideas!
#Calculus #HarmonicSeries #InfiniteSeries #Divergenceproofs #Beautyofanalysis

Note by Abhinav Raichur
6 years 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} ai−1 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 sin⁡θ \sin \theta
\boxed{123} 123 \boxed{123}

Comments

There are no comments in this discussion.

×

Problem Loading...

Note Loading...

Set Loading...