Alternating Series Test — Question 1

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

1. Verify every Alternating Series Test hypothesis.

2. Test absolute convergence and classify the series.

3. State its familiar value and give the error bound after $N$ terms.

*Question 1 – Solution

Write $b_n=1/n$. Then $b_n>0$, $b_{n+1}<b_n$, and $b_n\to0$. Hence the series converges by the Alternating Series Test.

$$\sum\left|\frac{(-1)^{n-1}}n\right|=\sum\frac1n,$$ which diverges. Therefore convergence is conditional.

The power series for $\ln(1+x)$ at $x=1$ gives $S=\ln2$. The Alternating Series Estimation Theorem gives $$|S-s_N|\le b_{N+1}=\frac1{N+1}.$$ Odd and even partial sums approach $\ln2$ from opposite sides.