Computing and Data Science

Sequent Sum

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  1. Complete the spreadsheets tutorial.
  2. Find the first five terms in the following sequences.
    1. \[a_n = \frac{2}{3n} \quad \quad n \geq 1 \]
    2. \[a_n = 2n^3 \quad \quad n \geq 0 \]
    3. \[a_1 = 5 \quad \quad a_{n+1} = 2a_{n}+3 \]
    4. \[a_0 = 1, a_1 = 2 \quad \quad a_{n} = 2 a_{n-1} + a_{n-2} \]
  3. Given the sequence: \[ \frac{7}{10}, \frac{8}{12}, \frac{9}{14}, \frac{10}{16}, ... \]
    1. Write the next three terms.
    2. Write a definition of the sequence.
    3. Find the limit of the sequence.
    4. Is the sum of this infinite sequence infinite?
  4. Given the sequence: \[ \frac{2}{2}, \frac{4}{3}, \frac{6}{5}, \frac{8}{7}, \frac{10}{11}, \frac{12}{13}, \frac{14}{17}, ... \]
    1. Write the next three terms.
    2. Write a definition of the sequence.
    3. Find the limit of the sequence.
    4. Is the sum of this infinite sequence infinite?
  5. The Fibonacci sequence is defined as: \[ F_1 = 1, \quad F_2 = 1\] \[F_{n} = F_{n-1} + F_{n-2} \]
    1. Write the first ten Fibonacci numbers.
    2. Write the first ten reciprocals of Fibonacci numbers, i.e., \(\frac{1}{F_1},\frac{1}{F_2},\frac{1}{F_3}, \dots \)
    3. Find (or approximate): \[\sum_{i=1}^{\infty} \frac{1}{F_i}\]
  6. Both of the following infinite sums converge. For each, expand the first ten terms of the sum and approximate the infinite sum by taking the partial sum of those ten terms. What value does each sum appear to be converging toward? Okay, you may want to use more than ten terms.
    \[\sum_{n=1}^{\infty} \frac{1}{n^2}\]\[\quad\quad\quad\]\[\sum_{n=1}^{\infty} \frac{1}{n^3}\]
  7. The infinite sum \[\sum_{n=1}^{\infty} \frac{1}{n^k}\] converges when \(k>1\). Which value is larger, \[\sum_{n=1}^{\infty} \frac{1}{n^{435}} \quad \quad \text{ or } \quad \quad \sum_{n=1}^{\infty} \frac{1}{n^{436}}\] and what can you conjecture (or prove!) about infite sums of this form?
  8. The \(n^{th}\) triangle number, \(T(n)\), is given by: \[T(n) = \sum_{k=1}^{n} k = \frac{n(n+1)}{2} \] Let the TriTri number \(TT(n)\) be the sum of the first \(n\) triangle numbers. So we would have \(TT(4) = 1+3+6+10=20\).
    1. Write an expression for \(TT(n)\) using \(\Sigma\)-notation.
    2. Find a closed formula for \(TT(n)\).
    3. Let \(TTT(n)\) be the TriTriTri number given by the sum of the first \(n\) TriTri numbers. What's the deal with \(TTT(n)\)?
    4. Dare I ask, what's up with \(TTTT(n)\)?