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:

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  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 n\ge 0.}

Step 2:
Also,
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}b_n=\lim_{n\rightarrow \infty}\frac{1}{n+1}=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 (-1)^n\frac{1}{n+1}}   converges
by the Alternating Series Test.

Final Answer:
(a)
(b)    converges

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