## Try It Now

Work these exercises to see how well you understand this material.

### Exercises

1. Calculate the following series:
1. $\sum_{i=1}^{3}(2 + 3i)$
2. $\sum_{i=-2}^{1} i^2$
3. $\sum_{j=0}^{n} 2^j$ for n = 1, 2, 3, 4
4. $\sum_{k=1}^{n}(2k - 1)$ for n = 1, 2, 3, 4

1. Express the formula $\sum_{i=1}^{n} \frac{1}{i(i+1)} = \frac{n}{n+1}$ without using summation notation.
2. Verify this formula for n = 3.
3. Repeat parts (a) and (b) for $\sum_{i=1}^{n} i^3 = \frac{n^2(n+1)^2}{4}$

2. Rewrite the following without summation sign for n = 3. It is not necessary that you understand or expand the notation $\binom{n}{k}$ at this point. $(x+y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k}y^k$.

3. For any positive integer k, let Ak = {x ∈ Q : k − 1 < x ≤ k} and Bk = {x ∈ Q : −k < x < k}. What are the following sets?
1. $\cup_{i=1}^{5} A_i$
2. $\cup_{i=1}^{5} B_i$
3. $\cap_{i=1}^{5} A_i$
4. $\cap_{i=1}^{5} B_i$

4. The symbol $\Pi$ is used for the product of numbers in the same way that $\Sigma$ is used for sums. For example, $\prod_{i=1}^5 x_i = x_1 x_2 x_3 x_4 x_5$. Evaluate the following:
1. $\prod_{i=1}^{3} i^2$
2. $\prod_{i=1}^{3} (2i + 1)$