009C Sample Midterm 1, Problem 2 Detailed Solution

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Consider the infinite series  

(a) Find an expression for the  th partial sum    of the series.

(b) Compute  

Background Information:  
The  th partial sum,    for a series    is defined as




Step 1:  
We need to find a pattern for the partial sums in order to find a formula.
We start by calculating    We have
Step 2:  
Next, we calculate    and    We have
Step 3:  
If we look at    and    we notice a pattern.
From this pattern, we get the formula


Step 1:  
From Part (a), we have
Step 2:  
We now calculate  
We get

Final Answer:  

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