Alternating Series Test — Question 7

*Question 7 Consider $\displaystyle\sum_{n=1}^{\infty}(-1)^ne^{-n}$.

1. Verify the AST hypotheses.

2. Test absolute convergence using a geometric series.

3. Find the exact sum and classify convergence.

*Question 7 – Solution

With $b_n=e^{-n}>0$, we have $b_{n+1}=e^{-1}b_n<b_n$ and $b_n\to0$. Thus the series converges.

$$\sum|(-1)^ne^{-n}|=\sum(e^{-1})^n,$$ a geometric series with ratio $e^{-1}<1$. Hence convergence is absolute.

The original series is geometric with first term $-e^{-1}$ and ratio $-e^{-1}$: $$S=\frac{-e^{-1}}{1+e^{-1}}=-\frac1{e+1}.$$