Skip to main content
\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)
Mathematics LibreTexts

4.E: Convergence of Sequences and Series (Exercises)

 

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)

Q1

Prove that if \(\lim_{n \to \infty }s_n = s\) then \(\lim_{n \to \infty }\left |s_n \right | = \left |s \right |\). Prove that the converse is true when \(s = 0\), but it is not necessarily true otherwise.

Q2

  1. Let \((s_n)\) and \((t_n)\) be sequences with \(s_n ≤ t_n,∀n\). Suppose \(\lim_{n \to \infty }s_n = s\) and \(\lim_{n \to \infty }t_n = t\). Prove \(s ≤ t\). [Hint: Assume for contradiction, that \(s > t\) and use the definition of convergence with \(ε = \(frac{s-t}{2}\) to produce an \(n\) with \(s_n > t_n\).]
  2. Prove that if a sequence converges, then its limit is unique. That is, prove that if \(\lim_{n \to \infty }s_n = s\) and \(\lim_{n \to \infty }s_n = s\), then \(s = t\).

Q3

Prove that if the sequence \((s_n)\) is bounded then \(\lim_{n \to \infty }\left (\frac{s_n}{n} \right ) = 0\).

Q4

  1. Prove that if \(x \neq 1\), then \[1 + x + x^2 +\cdots + x^n = \frac{1 - x^{n+1}}{1-x}\]
  2. Use (a) to prove that if \(|x| < 1\), then \(\lim_{n \to \infty }\left ( \sum_{j=0}^{n} x^j \right ) = \frac{1}{1-x}\)

Q5

Prove \[\lim_{n \to \infty }\frac{a_0 + a_1n + a_2n^2 +\cdots + a_kn^k}{b_0 + b_1n + b_2n^2 +\cdots + b_kn^k} = \frac{a_k}{b_k}\]

provided \(b_k \neq 0\). [Notice that since a polynomial only has finitely many roots, then the denominator will be non-zero when n is sufficiently large.]

Q6

Prove that if \(\lim_{n \to \infty }s_n = s\) and \(\lim_{n \to \infty }(s_n - t_n) = 0\), then \(\lim_{n \to \infty }t_n = s\).

Q7

  1. Prove that if \(\lim_{n \to \infty }s_n = s\) and \(s < t\), then there exists a real number \(N\) such that if \(n > N\) then \(s_n < t\).
  2. Prove that if \(\lim_{n \to \infty }s_n = s\) and \(r < s\), then there exists a real number \(M\) such that if \(n > M\) then \(r < s_n\).

Q8

Suppose \((s_n)\) is a sequence of positive numbers such that \(\lim_{n \to \infty }\left ( \frac{s_{n+1}}{s_n} \right ) = L\)

  1. Prove that if \(L < 1\), then \(\lim_{n \to \infty }s_n = 0\). [Hint: Choose \(R\) with \(L < R < 1\). By the previous problem, \(∃\; N\) such that if \(n > N\), then \(\frac{s_{n+1}}{s_n} < R\). Let \(n_0 > N\) be fixed and show \(s_{n_0+k} < R^ks_{n_0}\). Conclude that \(\lim_{k \to \infty }s_{n_0+k} = 0\) and let \(n = n_0 + k\).]
  2. Let \(c\) be a positive real number. Prove \[\lim_{n \to \infty }\left ( \frac{c^n}{n!} \right ) = 0\]

Contributor

  • Eugene Boman (Pennsylvania State University) and Robert Rogers (SUNY Fredonia)