REFER TO IMAGE FOR SET UP Consider the harmonic series: ∑ n = 1 ∞ 1 n Draw a graph of the function f ( x ) = 1 x on the domain [ 1 , ∞ ) . In order to use the Integral Test, we need to check that our...

REFER TO IMAGE FOR SET UP

Consider the harmonic series: ∑ n = 1 ∞ 1 n

1. Draw a graph of the function f ( x ) = 1 x  on the domain [ 1 , ∞ ) .


2. In order to use the Integral Test, we need to check that our function is continuous, positive, and decreasing. Explain how we know these things for this problem.


3. On your graph of the function f ( x ) , draw a box of height 1 over the x-interval [ 1 , 2 ] , then a box of height 1/2 over the x-interval [ 2 , 3 ], and so on. Your boxes will touch the curve at their upper left corners. This is a left endpoint Riemann sum for ∫ 1 ∞ f ( x ) d x  .


4. Explain the relationship between these four quantities. Are they greater, less than, or equal to each other? How do you know this?
   
   
   
   
   • ∑ n = 1 ∞ 1 n
   
   
   • The area under the curve y = f ( x )  from x=1 out to infinity
   
   
   • The total area of all of the boxes (out to infinity)
   
   
   • The value of the integral ∫ 1 ∞ f ( x ) d x
   
   
   
   


5. Evaluate the improper integral ∫ 1 ∞ 1 x d x . Does the improper integral converge or diverge?


6. Draw a conclusion about the convergence of the harmonic series. Does the series converge or diverge?

Extracted text: 1. Consider the harmonic series: n-1 1 a. Draw a graph of the function f(x) b. In order to use the Integral Test, we need to check that our function is continuous, positive, and decreasing. Explain how we know these things for this - on the domain [1, 0) . problem. c. On your graph of the function f(x), draw a box of height 1 over the x-interval [1, 2], then a box of height 1/2 over the x-interval [2, 3], and so on. Your boxes will touch the curve at their upper left corners. This is a left endpoint Riemann sum for f(x) dx . d. Explain the relationship between these four quantities. Are they greater, less than, or equal to each other? How do you know this? · The area under the curve y = f(x) from x=1 out to infinity · The total area of all of the boxes (out to infinity) - The value of the integral f(x) dx e. Evaluate the improper integral dx . Does the improper integral converge or diverge? f. Draw a conclusion about the convergence of the harmonic series. Does the series converge or diverge?