Waste less time on Facebook — follow Brilliant.
×

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 \[ I(a,x) = \int_0^{x} \frac{(\sin x)^{a}}{\cos x} dx \] we will tease this until it cries out an interesting series .....

multiplying 1 continuously OR \[ (\sin x)^2 + (\cos x)^2 \] yields \[ = \int_0^{x} (\frac{(\sin x)^{a}}{\cos x})((\sin x)^2 + (\cos x)^2) ...... dx \] \[ = \int_0^{x} (\sin x)^{a}\cos x + (\frac{(\sin x)^{a+2}}{\cos x})((\sin x)^2 + (\cos x)^2) dx \] \[ = \int_0^{x} ( (\sin x)^{a}\cos x + (\sin x)^{a+2}\cos x + (\sin x)^{a+4}\cos x ....... \infty ) dx \] \[ = ( \frac{(\sin x)^{a+1}}{a+1} + \frac{(\sin x)^{a+3}}{a+3} + \frac{(\sin x)^{a+5}}{a+5} ..... \infty ) \] plug in \[ a=1 ; x = \frac{\pi}{2} \] \[ = \frac{1}{a+1} + \frac{1}{a+3} + \frac{1}{a+5} ...... \infty \] to get \[ = \frac{1}{2} ( \frac{1}{1} + \frac{1}{2} + \frac{1}{3} ...... \infty ) \] which is half the harmonic series ........ solving the integral for the same a=1 gives \[ I(1,\frac{\pi}{2}) = \int_0^{\frac{\pi}{2}} \tan x dx \] \[ = \infty \] ....... 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!

Note by Abhinav Raichur
2 years, 6 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

There are no comments in this discussion.

×

Problem Loading...

Note Loading...

Set Loading...