Absolute Convergence and Divergence — Question 9

Question 9 – Solution

Step 1: Find the magnitude limit.

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

Step 2: Examine the signed terms.

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

Step 3: Apply the nth-term test.

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