Respuesta :

LammettHash
Recall that [tex]|\sin x|\le1[/tex], so this is an alternating series.

The series will then converge iff [tex]\left|\dfrac{\sin na}{\ln10^n}\right|\to0[/tex] as [tex]n\to\infty[/tex] and this summand is non-increasing.

You have

[tex]\left|\dfrac{\sin na}{\ln10^n}\right|\le\dfrac1{\ln10^n}=\dfrac1{n\ln10}\to0[/tex]

and [tex]\dfrac1{n\ln10}[/tex] is clearly strictly decreasing. This means the alternating series converges.

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