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 \(\sum_{i=1}^{n} i^3 = \frac{n^2(n+1)^2}{4}\)

  3. 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. \((x+y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k}y^k\).

  4. 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. \(\cap_{i=1}^{5} B_i\)

  5. 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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