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# Does the trigonometric series converge?

Examine if the series

converges.

Solution

The key lies in the fact that the number

is an integer. Let and let be the distance from to the nearest integer. That is

It is immediate that can be expanded periodically with period . Since is an integer we can get that

Thus:

and the conclusion follows.

1. John Khan says:

Regrouping terms (even and odd) we see that the odd ones cancel the root. What remains is actually an integer since the binomial coefficient is an integer.

• John Khan says:

Hey T,

what about examining the convergence of this series?

• Hi John,

sorry for replying after one year and so. The series converges. Indeed,

and the result follows. Is there any way to evaluate it in a closed form?

2. There appears to be more in this series than meets the eyes. The convergence of this series is not exceptional! It is known that for almost all (i.e except a set of Lebesgue measure ),
, the fractional part of is an equidistributed sequence. A consequence of this is for
almost all , the sequence does not converge to and hence the series diverges.

There are known exceptions to this. In particular, it is known that is not equidistributed mod 1 if is a PV number, i.e. an algebraic integer and all other roots of its minimal polynomials lie strictly inside the unit circle.

Hence the sum converges whenever is a PV number. Since is a PV number, the corresponding series do converge.