Absolute Convergence and Divergence — Question 9

*Question 9 Consider $\displaystyle\sum_{n=1}^{\infty}(-1)^n\left(1+\frac1n\right)^{-n}$.

1. Find the limit of the magnitude.

2. Apply the nth-term test.

3. Explain why alternating signs cannot rescue the series.

*Question 9 – Solution

The standard exponential limit gives $$\left(1+\frac1n\right)^n\to e, \qquad \left(1+\frac1n\right)^{-n}\to e^{-1}\ne0.$$

Even-indexed terms approach $e^{-1}$ and odd-indexed terms approach $-e^{-1}$. Thus the terms do not approach zero.

A necessary condition for series convergence fails, so the series diverges. The AST is unavailable because its magnitude condition $b_n\to0$ fails.