Friday, 11 December 2015

calculus - Show by comparison that sumlimitsin=1nftysin(frac1n) diverges?




By using the integral test, I know that n=1sin(1n) diverges. However, how would I show that the series diverges using the limit comparison test? Would I simply let n=1an=n=1bn=n=1sin(1n) and then take lim to show the series diverges (assuming the limit converges to some nonnegative, finite value)?


Answer



Notice x_n = \frac{1}{n} , then \sum \frac{1}{n}, the harmonic series, we all know is divergent. Now,



\lim \frac{ \sin (\frac{1}{n})}{\frac{1}{n}} =_{t = \frac{1}{n}} \lim_{t \to 0} \frac{ \sin t}{t} = 1



The result now follows by the limit comparison test. :)


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