3.1 Sums & Products

Let \(f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+} \), \(S ::= \sum\limits_{i=1}^{n} f(i) \), \(I ::= \int\limits_{1}^{n} f(x) dx \).

1. What is the upper bound for \(S \) when \(f \) is weakly increasing?

   Exercise 1

     \(I \) \(I + f(1) \) \(I + f(n) \) \(ln(n) \) \(I + (1/2)f(n) \) \(I + f(n) \) 

2. What is the upper bound for \(S \) when \(f \) is weakly decreasing?

   Exercise 2

     \(I \) \(I + f(1) \) \(I + f(n) \) \(ln(n) \) \(I + (1/2)f(n) \) \(I + f(1) \) 

3. What is the lower bound for \(S \) when \(f \) is weakly increasing?

   Exercise 3

     \(I \) \(I + f(1) \) \(I + f(n) \) \(ln(n) \) \(I + (1/2)f(n) \) \(I + f(1) \) 

4. What is the lower bound for \(S \) when \(f \) is weakly decreasing?

   Exercise 4

     \(I \) \(I + f(1) \) \(I + f(n) \) \(ln(n) \) \(I + (1/2)f(n) \) \(I + f(n) \) 

5. Do the upper bounds and lower bounds for \(f \) change if it is strictly increasing/decreasing instead of weakly increasing/decreasing?

   Exercise 5

     yes no it depends no 
   From weakly increasing/decreasing to strictly increasing/decreasing, simply change the inequality sign from \(\le \) to <.

6. What is the upper bound for \(H_n \)?

   Exercise 6

   &nbsp; \(1 + ln(n) \) \(ln(n+1) \) \(ln(n) \) \(1 + ln(n) \)&nbsp;

7. What is the lower bound for \(H_n \)?

   Exercise 7

   &nbsp; \(1 + ln(n) \) \(ln(n+1) \) \(ln(n) \) \(ln(n+1) \)&nbsp;

8. What is the asymptotic bound for \(H_n \)?

   Exercise 8

   &nbsp; \(1 + ln(n) \) \(ln(n+1) \) \(ln(n) \) \(ln(n) \)&nbsp;
   The integral method is for finding the upper and lower bounds of a sum, but it is also helpful in obtaining the asymptotic bound / asymptotic equivalence / asymptotic equality of the sum. Once we have the upper and lower bounds, we take the limit on these bounds as \(n \rightarrow \infty \).
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