Alternating Series Test — Question 9

*Question 9 Consider $\displaystyle\sum_{n=1}^{\infty}(-1)^n\sin(1/n)$.

1. Verify the AST hypotheses using monotonicity of sine.

2. Test absolute convergence by limit comparison with $1/n$.

3. Classify the series and state the alternating error bound.

*Question 9 – Solution

For $n\ge1$, $1/n\in(0,1]$ and $b_n=\sin(1/n)>0$. Since $1/(n+1)<1/n$ and sine is increasing on $[0,1]$, $b_{n+1}<b_n$. Continuity gives $b_n\to\sin0=0$. Thus the series converges.

$$\lim_{n\to\infty}\frac{\sin(1/n)}{1/n} =\lim_{x\to0}\frac{\sin x}{x}=1.$$ The harmonic series diverges, so Limit Comparison shows $\sum\sin(1/n)$ diverges. Hence convergence is conditional.

$$|S-s_N|\le\sin\left(\frac1{N+1}\right)<\frac1{N+1}.$$