# 3.1 Sums & Products

## Integral Method Demystified

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
2. What is the upper bound for $$S$$ when $$f$$ is weakly decreasing?

Exercise 2
3. What is the lower bound for $$S$$ when $$f$$ is weakly increasing?

Exercise 3
4. What is the lower bound for $$S$$ when $$f$$ is weakly decreasing?

Exercise 4
5. Do the upper bounds and lower bounds for $$f$$ change if it is strictly increasing/decreasing instead of weakly increasing/decreasing?

Exercise 5
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
7. What is the lower bound for $$H_n$$?

Exercise 7
8. What is the asymptotic bound for $$H_n$$?

Exercise 8
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 $$n \rightarrow \infty$$.