009C Sample Final 2, Problem 3

Determine if the following series converges or diverges. Please give your reason(s).

(a)  $\sum _{n=0}^{+\infty }{\frac {n!}{(2n)!}}$

(b)  $\sum _{n=0}^{+\infty }(-1)^{n}{\frac {1}{n+1}}$

Foundations:
1. 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  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 L=1,}   the test is inconclusive.
2. If a series absolutely converges, then it also converges.
3. Alternating Series Test
Let  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 \{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.

Solution:

(a)

Step 1:

Step 2:

(b)

Step 1:
For
$\sum _{n=1}^{\infty }(-1)^{n}{\frac {1}{n+1}},$
we notice that this series is alternating.
Let  $b_{n}={\frac {1}{n+1}}.$
The sequence  $\{b_{n}\}$  is decreasing since
${\frac {1}{n+2}}<{\frac {1}{n+1}}$
for all  $n\geq 0.$

Step 2:
Also,
$\lim _{n\rightarrow \infty }b_{n}=\lim _{n\rightarrow \infty }{\frac {1}{n+1}}=0.$
Therefore, the series  $\sum _{n=1}^{\infty }(-1)^{n}{\frac {1}{n+1}}$   converges
by the Alternating Series Test.

Final Answer:
(a)
(b)    converges

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