We all know that sums of periodic function are divergent. Take a look at this series:
It is divergent yes, but look at the following manipulations:
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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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:
n=1∑∞ncosn
=ℜn=1∑∞nein
=−ℜln(1−ei)
=−2ln(1−ei)+ln(1−e−i)
=−21ln(2−2cos1)
Now this series Converges to this value(you can check by wolfram alpha) .we know that the sum ∑n=1∞nxn converges iff ∣x∣<1 . But I used the same x=ei used in the periodic sum. How can that be justified??
I mean if we say that ∣ei∣=1, this will contradict the convergence of ∑n=1∞ncosn.
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It is not true that ∑nxn converges iff ∣x∣<1.
The proper version of the statement is that ∑nxn converges absolutely iff ∣x∣<1.
For example, we know that ∑n(−1)n converges conditionally to ln2.
(I believe that) we get conditional convergence if we substitute ∣x∣=1,x=1. In particular, x=ei is valid.
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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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