009C Sample Midterm 2, Problem 3

Determine convergence or divergence:

a) $\sum _{n=1}^{\infty }(-1)^{n}{\sqrt {\frac {1}{n}}}$

b) $\sum _{n=1}^{\infty }(-2)^{n}{\frac {n!}{n^{n}}}$

Foundations:
Alternating Series Test
Ratio Test

Solution:

(a)

Step 1:
First, we have
$\sum _{n=1}^{\infty }(-1)^{n}{\sqrt {\frac {1}{n}}}=\sum _{n=1}^{\infty }(-1)^{n}{\frac {1}{\sqrt {n}}}.$

Step 2:
We notice that the series is alternating.
Let $b_{n}={\frac {1}{\sqrt {n}}}.$
The sequence $\{b_{n}\}$ is decreasing since
${\frac {1}{\sqrt {n+1}}}<{\frac {1}{\sqrt {n}}}$
for all $n\geq 1.$
Also,
$\lim _{n\rightarrow \infty }b_{n}=\lim _{n\rightarrow \infty }{\frac {1}{\sqrt {n}}}=0.$
Therefore, the series Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{n=1}^\infty \frac{(-1)^n}{\sqrt{n|}} converges by the Alternating Series Test.

(b)

Step 1:

Step 2:

Final Answer:
(a)     converges
(b)

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