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

       


Solution:

(a)

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
       
and
       
Step 3:  
If we look at    and    we notice a pattern.
From this pattern, we get the formula
       

(b)

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


Final Answer:  
    (a)    
    (b)    

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