Sequences and Series

MATH 116 SI worksheet · Fall 2025 · Session 3

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Definitions

A sequence of real numbers is a function \(f: \mathbb{N} \to \mathbb{R}\), \(n \mapsto a_n = f(n)\), written \((a_n)_{n \in \mathbb{N}} = (a_1, a_2, a_3, \dots)\).

Its associated series is \(\displaystyle \sum_{n=1}^\infty a_n = a_1 + a_2 + a_3 + \cdots\), and the sequence of partial sums is \(\displaystyle S_n = \sum_{k=1}^n a_k\).

The series is defined by \(\displaystyle \sum_{n=1}^\infty a_n = \lim_{n \to \infty} S_n\). It is

  • convergent if \(\lim_{n \to \infty} S_n = s\) for some \(s \in \mathbb{R}\), and
  • divergent if \(\lim_{n \to \infty} S_n = \pm\infty\) or the limit does not exist.

Necessary condition: if \(\sum a_n\) converges, then \(\lim_{n\to\infty} a_n=0\).

Write \(a_n\) for each sequence

  1. \(1, 2, 3, \dots\) Answer: \(a_n = n\)
  2. \[2, 4, 6, \dots\]
  3. \[1, 2, 4, 8, \dots\]
  4. \[1, 1, 2, 3, 5, 8, \dots\]
  5. \[1, 2, 6, 24, 120, \dots\]
  6. \[1, 3, 6, 10, 15, 21, \dots\]

Rewrite as a sum

  1. \(1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+ \cdots\) Answer: \(\sum_{n=0}^\infty\frac{1}{2^n}\)
  2. \[\frac{1}{1\cdot2}+\frac{1}{2\cdot3}+\frac{1}{3\cdot4}+ \cdots\]
  3. \[1+\frac{1}{4}+\frac{1}{9}+\frac{1}{16}+ \cdots\]
  4. \[1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+ \cdots\]
  5. \[1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+ \cdots\]
  6. \[1-1+1-1+ \cdots\]
  7. \[1-2+3-4+5- \cdots\]
  8. \[1+2+3+4+5+ \cdots\]

Properties of series

Assume all series below converge.

  • Addition: \(\sum (a_n + b_n) = \sum a_n + \sum b_n\)
  • Subtraction: \(\sum (a_n - b_n) = \sum a_n - \sum b_n\)
  • Scalar multiple: \(\sum c\,a_n = c \sum a_n\) for \(c \in \mathbb{R}\)

Compute the sums

  1. \[\displaystyle \sum_{n=1}^\infty \frac{3^{n-1}-1}{6^{n-1}}\]
  2. \[\displaystyle \sum_{n=1}^\infty \frac{4}{6^{n-1}}\]

Find a formula for the partial sums

  1. \[1+2+3+\cdots +n\]
  2. \[1^2+2^2+3^2+\cdots +n^2\]
  3. \[1^3+2^3+3^3+\cdots +n^3\]

Additional problems

  1. Find a formula for the general term of \(1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5, \dots\)
  2. Find a closed form for the general term of the sequence defined by \(x_0=1\) and \(x_n = x_{n-1} + n\) if \(n\) is odd, \(x_n = x_{n-1} + n - 1\) if \(n\) is even.
  3. Find the sum of the coefficients of the terms whose exponent of \(x\) is a multiple of 3 in \((1 + x^2 - x^3 + x^4)^{10}\).