Is the Series ( sum_{n1}^{infty} sinleft(frac{1}{n}right) ) Convergent or Divergent?

Is the Series ( sum_{n1}^{infty} sinleft(frac{1}{n}right) ) Convergent or Divergent? H1: Understanding the Convergence or Divergence of ( sum_{n1}^{infty} sinleft(frac{1}{n}right) )

To determine whether the series ( sum_{n1}^{infty} sinleft(frac{1}{n}right) ) converges or diverges, we must analyze the behavior of its terms ( sinleft(frac{1}{n}right) ) as ( n ) approaches infinity.

H2: Analyzing the Behavior of ( sinleft(frac{1}{n}right) )

As ( n ) becomes very large, ( frac{1}{n} ) approaches 0. We can utilize the Taylor series expansion for ( sin(x) ) around ( x 0 ).

Step 1: The Taylor series expansion for ( sin(x) ) around ( x 0 ) is given by:

( sin(x) approx x ) for sufficiently small ( x )

Thus, we have:

( sinleft(frac{1}{n}right) approx frac{1}{n} ) as ( n to infty )

H2: Comparison with a Known Series

The series ( sum_{n1}^{infty} frac{1}{n} ) is the harmonic series, which is known to diverge. Since ( sinleft(frac{1}{n}right) ) behaves like ( frac{1}{n} ) for large ( n ), we can employ the Limit Comparison Test to draw a conclusion.

H3: Applying the Limit Comparison Test

Let’s compare ( sinleft(frac{1}{n}right) ) with ( frac{1}{n} ) using the Limit Comparison Test:

( lim_{n to infty} frac{sinleft(frac{1}{n}right)}{frac{1}{n}} lim_{n to infty} n cdot sinleft(frac{1}{n}right) )

Using the fact that ( sin(x) approx x ) for small ( x ), we obtain:

( lim_{n to infty} n cdot sinleft(frac{1}{n}right) lim_{n to infty} n cdot frac{1}{n} 1 )

Since this limit is a positive finite number, we can conclude that:

( sum_{n1}^{infty} sinleft(frac{1}{n}right) ) behaves like ( sum_{n1}^{infty} frac{1}{n} )

H4: Conclusion

Bearing in mind that the harmonic series ( sum_{n1}^{infty} frac{1}{n} ) diverges, the series ( sum_{n1}^{infty} sinleft(frac{1}{n}right) ) must also diverge.

Thus, the series ( sum_{n1}^{infty} sinleft(frac{1}{n}right) ) is divergent.