Prove that the series ∑i=1∞1nsin⁡(1n) is divergent

Ernest Ryland

Ernest Ryland

Answered question


Prove that the series i=11nsin(1n) is divergent

Answer & Explanation



Beginner2022-01-18Added 34 answers

For 0xπ2, we have
Therefore use 2πxsin(x),x=1n, and get after multiplication 2π1nsin(1n).
Paul Mitchell

Paul Mitchell

Beginner2022-01-19Added 40 answers

A variant is to use limu0sinuu=1 and in particular this term is >12 for |u|1 small enough.
Now write 1nsin(1n)=1n×sin(1n)1n>1n×12 for n large enough.
And still conclude by comparison with 1n
Note: a faster way is to use equivalents, the limit 1 means 1nsin(1n)1n

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