Alternating Series Test — Question 3

*Question 3 Consider $\displaystyle\sum_{n=1}^{\infty}(-1)^n\frac{n+1}{n}$.

1. Evaluate the limit of the term magnitude.

2. Apply the nth-term test before considering the AST.

3. Explain exactly which AST condition fails.

*Question 3 – Solution

$$b_n=\frac{n+1}{n}=1+\frac1n\longrightarrow1.$$ Thus the series terms $a_n=(-1)^nb_n$ alternate near $\pm1$ and do not approach zero.

Every convergent series must satisfy $a_n\to0$. Here the even subsequence tends to $1$ and the odd subsequence tends to $-1$, so the term sequence has no limit and the series diverges.

Although $b_n$ is positive and decreasing, the required condition $b_n\to0$ fails. Alternating signs cannot compensate for terms of persistent size.