Finding convergence/abs convergence/divergence of series with trig function ∑ n = 1 ∞ sin ⁡...

minwaardekn

minwaardekn

Answered

2022-07-01

Finding convergence/abs convergence/divergence of series with trig function
n = 1 sin ( 4 n ) 4 n

Answer & Explanation

svirajueh

svirajueh

Expert

2022-07-02Added 29 answers

We can note that
n 1 sin ( 4 n ) 4 n = Im ( n 1 ( e 4 i 4 ) n )
and since
| e 4 i 4 | = 1 4
| e 4 i 4 | = 1 4
we can calculate the sum
n 1 ( e 4 i 4 ) n = e 4 i 4 e 4 i
and so
Im ( n 1 ( e 4 i 4 ) n ) = Im ( e 4 i 4 e 4 i ) = 4 sin ( 4 ) 17 8 cos ( 4 ) 0.13619.

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