Alternating Series Test — Question 1

Question 1 – Solution

Step 1: Verify the AST hypotheses.

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.

Step 2: Test absolute convergence.

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

Step 3: State the value and error.

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.