009C Sample Midterm 1, Problem 2

Consider the infinite series $\sum _{n=2}^{\infty }2{\bigg (}{\frac {1}{2^{n}}}-{\frac {1}{2^{n+1}}}{\bigg )}.$

a) Find an expression for the $n$th partial sum $s_{n}$ of the series.

b) Compute $\lim _{n\rightarrow \infty }s_{n}.$

Foundations:
The $n$th partial sum, $s_{n}$ for a series $\sum _{n=1}^{\infty }a_{n}$
is defined as
$s_{n}=\sum _{i=1}^{n}a_{i}.$

Solution:

(a)

Step 1:
We need to find a pattern for the partial sums in order to find a formula.
We start by calculating $s_{2}$. We have
$s_{2}=2{\bigg (}{\frac {1}{2^{2}}}-{\frac {1}{2^{3}}}{\bigg )}.$

Step 2:
Next, we calculate $s_{3}$ and $s_{4}.$ We have

(b)

Step 1:

Step 2:

Final Answer:
(a)
(b)

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