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:
1. Alternating Series Test
Let $\{a_{n}\}$ be a positive, decreasing sequence where $\lim _{n\rightarrow \infty }a_{n}=0.$
Then, $\sum _{n=1}^{\infty }(-1)^{n}a_{n}$ and $\sum _{n=1}^{\infty }(-1)^{n+1}a_{n}$
converge.
2. Ratio Test
Let $\sum a_{n}$ be a series and $L=\lim _{n\rightarrow \infty }{\bigg |}{\frac {a_{n+1}}{a_{n}}}{\bigg |}.$
Then,
If $L<1,$ the series is absolutely convergent.
If $L>1,$ the series is divergent.
If $L=1,$ the test is inconclusive.
3. If a series absolutely converges, then it also converges.

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 $\sum _{n=1}^{\infty }{\frac {(-1)^{n}}{\sqrt {n}}}$ converges by the Alternating Series Test.

(b)

Step 1:
We begin by using the Ratio Test.
We have
${\begin{array}{rcl}\displaystyle {\lim _{n\rightarrow \infty }{\bigg |}{\frac {a_{n+1}}{a_{n}}}{\bigg |}}&=&\displaystyle {\lim _{n\rightarrow \infty }{\bigg |}{\frac {(-2)^{n+1}(n+1)!}{(n+1)^{n+1}}}{\frac {n^{n}}{(-2)^{n}n!}}{\bigg |}}\\&&\\&=&\displaystyle {\lim _{n\rightarrow \infty }{\bigg |}(-2)(n+1){\frac {n^{n}}{(n+1)^{n+1}}}{\bigg |}}\\&&\\&=&\displaystyle {\lim _{n\rightarrow \infty }2{\frac {n^{n}}{(n+1)^{n}}}}\\&&\\&=&\displaystyle {2\lim _{n\rightarrow \infty }{\bigg (}{\frac {n}{n+1}}{\bigg )}^{n}.}\end{array}}$

Step 2:
Now, we need to calculate $\lim _{n\rightarrow \infty }{\bigg (}{\frac {n}{n+1}}{\bigg )}^{n}.$
Let $y=\lim _{n\rightarrow \infty }{\bigg (}{\frac {n}{n+1}}{\bigg )}^{n}.$
Then, taking the natural log of both sides, we get
${\begin{array}{rcl}\displaystyle {\ln y}&=&\displaystyle {\ln {\bigg (}\lim _{n\rightarrow \infty }{\bigg (}{\frac {n}{n+1}}{\bigg )}^{n}{\bigg )}}\\&&\\&=&\displaystyle {\lim _{n\rightarrow \infty }\ln {\bigg (}{\frac {n}{n+1}}{\bigg )}^{n}}\\&&\\&=&\displaystyle {\lim _{n\rightarrow \infty }n\ln {\bigg (}{\frac {n}{n+1}}{\bigg )}}\\&&\\&=&\displaystyle {\lim _{n\rightarrow \infty }{\frac {\ln {\bigg (}{\frac {n}{n+1}}{\bigg )}}{\frac {1}{n}}}}\end{array}}$
since we can interchange limits and continuous functions.
Now, this limit has the form ${\frac {0}{0}}.$
Hence, we can use L'Hopital's Rule to calculate this limit.

Step 3:
Now, we have
${\begin{array}{rcl}\displaystyle {\ln y}&=&\displaystyle {\lim _{n\rightarrow \infty }{\frac {\ln {\bigg (}{\frac {n}{n+1}}{\bigg )}}{\frac {1}{n}}}}\\&&\\&=&\displaystyle {\lim _{x\rightarrow \infty }{\frac {\ln {\bigg (}{\frac {x}{x+1}}{\bigg )}}{\frac {1}{x}}}}\\&&\\&{\overset {L'H}{=}}&\displaystyle {\lim _{x\rightarrow \infty }{\frac {{\frac {1}{{\bigg (}{\frac {x}{x+1}}{\bigg )}}}{\frac {1}{(x+1)^{2}}}}{{\big (}-{\frac {1}{x^{2}}}{\big )}}}}\\&&\\&=&\displaystyle {\lim _{x\rightarrow \infty }{\frac {-x}{x+1}}}\\&&\\&=&\displaystyle {-1.}\end{array}}$

Step 4:
Since $\ln y=-1,$ we know
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 y=e^{-1}.}
Now, we have
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 \lim_{n\rightarrow \infty} \bigg|\frac{a_{n+1}}{a_n}\bigg|=2e^{-1}=\frac{2}{e}.}
Since 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 \frac{2}{e}<1,} the series is absolutely convergent by the Ratio Test.
Therefore, the series converges.

Final Answer:
(a)     converges
(b)     converges

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