Sigma Summation Notation

  1. Without using a computer or calculator or Faulhaber’s formulas, compute the value of each of these sums.

    \( \displaystyle \sum_{i=1}^{4} \tfrac{1}{i} \)
    \( \displaystyle \sum_{i=1}^{135} 2 \)
    \( \displaystyle \sum_{i=0}^{5} \Bigl(\tfrac{2}{3}i+2\Bigr) \)
    \( \displaystyle \sum_{i=1}^{9} (-1)^i \)
    \( \displaystyle \sum_{i=-9}^{10} i^3 \)

  2. Express each of the following sums in sigma summation notation.

    \(2 + 4 + 6 + 8 + 10 + 12\)
    \(1 + 3 + 5 + 7 + 9\)
    \(\frac{2}{3}+\frac{4}{9}+\frac{8}{27}+\frac{16}{81}\)
    \(6 - 7 + 8 - 9 + 10 - 11 + 12 - 13\)
    \(1 + 4 + 9 + 16 + \dotsb + 529\)
    \(\frac{\sqrt{3}}{2} + \frac{1}{2} + 0 - \frac{1}{2} - \frac{\sqrt{3}}{2} - 1 - \frac{\sqrt{3}}{2} - \frac{1}{2} - 0 + \frac{1}{2} + \frac{\sqrt{3}}{2} +1 \)

  3. Use Faulhaber’s formulas to compute the value of each of these sums. So that you may find confidence your answer, know that each sum consists only of the digits \(\{0,1,2,3\}\)

    \( \displaystyle \sum_{i=1}^{81} i \)
    \( \displaystyle \sum_{i=1}^{26} i^3 \)
    \( \displaystyle \sum_{i=1}^{1826} i^2 \)
    \( \displaystyle \sum_{i=1}^{99} \bigl(i-1\bigr)\bigl(i+2\bigr) \)

  4. First, for \(a \lt b,\) convince yourself of the following fact about summations whose index doesn’t start at zero. \[ \sum_{i=a}^{b} f(i) = \sum_{i=1}^{b} f(i) -\sum_{i=1}^{a-1} f(i) \] Then use Faulhaber’s formulas to compute the value of each of these sums. So that you may find confidence your answer, know that all three sums are equal.

    \( \displaystyle \sum_{i=9}^{118} i \)
    \( \displaystyle \sum_{i=20}^{30} i^2 \)
    \( \displaystyle \sum_{i=9}^{13} i^3 \)