Dirichlet test states that if a_n is a decreasing sequence with \lim_{n o \infty}a_n=0 , and b_n is a sequence such that its partial sums B_m=\sum_{n=1}^m b_n is bounded, then the series \sum_{n=1}^{\infty} a_nb_n is converging.
We are going to test the convergence of \sum_{n=1}^{\infty}\frac{\sin n}{n} . Which of the following claims are True regarding the correct choices and statements about a_n and b_n for the Dirichlet Test?

Question

Dirichlet test states that if    a_n  is a decreasing sequence with   \lim_{n	o \infty}a_n=0 , and   b_n  is a sequence such that its partial sums   B_m=\sum_{n=1}^m b_n  is bounded, then the series    \sum_{n=1}^{\infty} a_nb_n  is converging.  
 We are going to test the convergence of   \sum_{n=1}^{\infty}\frac{\sin n}{n} . Which of the following claims are True regarding the correct choices and statements about   a_n  and   b_n  for the Dirichlet Test?

Accepted Answer

With   b_n=\sin n  the partial sums  B_m=\sum_{n=1}^m \sin n=\sum_{n=1}^m \frac{\cos(n-1)-\cos(n+1)}{2\sin 1}\=\frac{1}{2 \sin 1}(\sum_{n=0}^{m-1} \cos n-\sum_{n=2}^{m+1} \cos n)\=\frac{1}{2 \sin 1}(\cos 0+\cos 1-\cos m-\cos (m+1)) is bounded.

Accepted Answer

With   a_n=\frac{1}{n} ,   a_n  is decreasing and   \lim_{n	o \infty}a_n=0 .

Answer

With   b_n=\frac{1}{n}  the partial sums   B_m=\sum_{n=1}^m \frac{1}{n}  is bounded since   m  is finite.

Answer

With   b_n=\sin n  the partial sums   |B_m|=|\sum_{n=1}^m \sin n|\leq \sum_{n=1}^m |\sin n|\leq \sum_{n=1}^m 1=m  is bounded.

Answer

With   a_n=\sin n ,   a_n  is decreasing and   \lim_{n	o \infty}a_n=0 .

Answer

With   b_n=\sin n  the partial sums   B_m=\sum_{n=1}^m \sin n  is bounded since   m  is finite.

Answer

With   b_n=\frac{1}{n}  the partial sums   B_m=\sum_{n=1}^m \frac{1}{n}\leq \sum_{n=1}^m 1=m  is bounded.

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Dirichlet test states that if a_n is a decreasing sequence with \lim_{n\to...

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