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)


Source: Al Doerr and Ken Levasseur, http://faculty.uml.edu/klevasseur/ads-latex/ads.pdf
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