Divergence of Infinite Series of Periodic Functions

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

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

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

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

=11ei\displaystyle = \Re \frac{1}{1-e^i}

=11ei+11ei2\displaystyle = \frac{\frac{1}{1-e^i} +\frac{1}{1-e^{-i}} }{2}

=12\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??

Note by Hasan Kassim
5 years, 10 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](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} 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}


Sort by:

Top Newest

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

Sudeep Salgia - 5 years, 10 months ago

Log in to reply

Now look at this:

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

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

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

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

=12ln(22cos1)\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 n=1xnn \sum_{n=1}^{\infty} \frac{x^n}{n} converges iff x<1 |x|<1 . But I used the same x=ei x = e^i used in the periodic sum. How can that be justified??

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

Hasan Kassim - 5 years, 10 months ago

Log in to reply

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

For example, we know that (1)nn \sum \frac{ (-1)^n} { n} converges conditionally to ln2 \ln 2 .

(I believe that) we get conditional convergence if we substitute x=1,x1 |x| = 1, x \neq 1 . In particular, x=ei x = e^i is valid.

Calvin Lin Staff - 5 years, 10 months ago

Log in to reply

@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 - 5 years, 10 months ago

Log in to reply

Log in to reply


Problem Loading...

Note Loading...

Set Loading...