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??
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Hasan Kassim
3 years, 1 month ago
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Top NewestThis 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.
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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 \(\sum_{n=1}^{\infty} \frac{\cos n}{n} \).
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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.
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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?
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@Calvin Lin @Michael Mendrin @Brian Charlesworth @Aman Raimann @Krishna Sharma @Ishan Dasgupta Samarendra
Any Thoughts??
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