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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