Divergence of Infinite Series of Periodic Functions

We all know that sums of periodic function are divergent. Take a look at this series:

$\displaystyle S=\sum_{n=0}^{\infty} \cos n$

It is divergent yes, but look at the following manipulations:

$\displaystyle = \Re \sum_{n=0}^{\infty} e^{in}$

$\displaystyle = \Re \frac{1}{1-e^i}$

$\displaystyle = \frac{\frac{1}{1-e^i} +\frac{1}{1-e^{-i}} }{2}$

$\displaystyle = \frac{1}{2}$

Shouldn't there be a mathematical error in some of the steps?? Analytically, the series should diverge since the summand is periodic and bounded, but what about the calculations??

#Calculus #brillianthelp #InfiniteSeries #Periodicfunctions

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.

Markdown	Appears as
`italics` or `_italics_`	italics
`bold` or `__bold__`	bold
- bulleted - list	bulleted list
1. numbered 2. list	numbered 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"

Math	Appears 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

This is because the geometric series will converge for $|x| <1$ . The fact that you have assumed convergence and applied the formula is causing the anomaly.

Sudeep Salgia - 4 years, 2 months ago

Now look at this:

$\displaystyle \sum_{n=1}^{\infty} \frac{\cos n}{n}$

$\displaystyle = \Re \sum_{n=1}^{\infty} \frac{e^{in}}{n}$

$\displaystyle = -\Re \ln (1-e^{i})$

$\displaystyle = -\frac{\ln (1-e^{i}) +\ln (1-e^{-i}) }{2}$

$\displaystyle = -\frac{1}{2} \ln (2-2\cos 1)$

Now this series Converges to this value(you can check by wolfram alpha) .we know that the sum $\sum_{n=1}^{\infty} \frac{x^n}{n}$ converges iff $|x|<1$ . But I used the same $x = e^i$ used in the periodic sum. How can that be justified??

I mean if we say that $|e^i| = 1$ , this will contradict the convergence of $\sum_{n=1}^{\infty} \frac{\cos n}{n}$ .

Hasan Kassim - 4 years, 2 months ago

It is not true that $\sum \frac { x^n } { n }$ converges iff $|x| < 1$ .
The proper version of the statement is that $\sum \frac { x^n } { n }$ converges absolutely iff $|x| < 1$ .

For example, we know that $\sum \frac{ (-1)^n} { n}$ converges conditionally to $\ln 2$ .

(I believe that) we get conditional convergence if we substitute $|x| = 1, x \neq 1$ . In particular, $x = e^i$ is valid.

Calvin Lin Staff - 4 years, 2 months ago

@Calvin Lin – Okay I got it.Thanks for the insight :)

Do you know a regularization to this series? "Assigning values to divergent series"?

And in General, for what reasons we assign values to divergent series? and doesn't that make any contradiction?

Hasan Kassim - 4 years, 2 months ago

@Calvin Lin @Michael Mendrin @Brian Charlesworth @Aman Raimann @Krishna Sharma @Ishan Dasgupta Samarendra

Any Thoughts??

Hasan Kassim - 4 years, 2 months ago

Math	Appears 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}`	$\boxed{123}$