Try It Now

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
      
      
3. 
   
   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 \(\sum_{i=1}^{n} i^3 = \frac{n^2(n+1)^2}{4}\)
      
      
5. 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. \((x+y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k}y^k\).
   
   
7. For any positive integer k, let A_k = {x ∈ Q : k − 1 < x ≤ k} and B_k = {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. \(\cap_{i=1}^{5} B_i\)
      
      
9. 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
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License.

1. Answer:
   1. 24
   2. 6
   3. 3, 7, 15, 31
   4. 1, 4, 9, 16
      
      
3. Answer:
   1. \(\frac{1}{1(1+1)} + \frac{1}{2(2+1)} + \frac{1}{3(3+1)} + ... + \frac{1}{n(n+1)} = \frac{n}{n+1}\)
   2. \(\frac{1}{1(2)} + \frac{1}{2(3)} + \frac{1}{3(4)} = \frac{1}{2} + \frac{1}{6} + \frac{1}{12} = \frac{3}{4} = \frac{3}{3+1}\)
   3. \(1+2^3+3^3 + ... + n^3 = (\frac{1}{4})n^2(n+1)^2 \; \; \; 1+8+27=36=(\frac{1}{4})(3)^2(3+1)^2\)
      
      
5. Answer: \((x+y)^3 = \binom{3}{0}x^3 + \binom{3}{1}x^2y + \binom{3}{2}xy^2 + \binom{3}{3}y^n\)
   
   
7. Answer:
   
   1. {x ∈ Q | 0 < x ≤ 5}
   2. {x ∈ Q | −5 < x < 5} = B⁵
   3. ∅
   4. {x ∈ Q | −1 < x < 1} = B₁
      
      
9. Answer:
   
   1. 36
   2. 105