009C Sample Midterm 2, Problem 1

Evaluate:

a) $\lim _{n\rightarrow \infty }{\frac {1}{{\big (}{\frac {n-4}{n}}{\big )}^{n}}}$

b) $\sum _{n=1}^{\infty }{\frac {1}{2}}{\bigg (}{\frac {1}{4}}{\bigg )}^{n-1}$

Foundations:
Sum formula for geometric series

Solution:

(a)

Step 1:

Step 2:

(b)

Step 1:
First, we not that this is a geometric series with $r={\frac {1}{4}}.$
Since $|r|={\frac {1}{4}}<1,$
this series converges.

Step 2:
Now, we need to find the sum of this series.
The first term of the series is $a_{1}={\frac {1}{2}}.$
Hence, the sum of the series is
${\begin{array}{rcl}\displaystyle {\frac {a_{1}}{1-r}}&=&\displaystyle {\frac {\frac {1}{2}}{1-{\frac {1}{4}}}}\\&&\\&=&\displaystyle {\frac {{\big (}{\frac {1}{2}}{\big )}}{{\big (}{\frac {3}{4}}{\big )}}}\\&&\\&=&\displaystyle {\frac {2}{3}}\end{array}}$

Final Answer:
(a)
(b)     ${\frac {2}{3}}$

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