Sequences

The Essentials

One type of sequence is an arithmetic sequence. Arithmetic sequences comprise of the same value being added to the previous term. This is an example:

\[ \begin{cases} a_0=2\\ a_n=a_{n-1}+3 \end{cases} \]
\[ \begin{equation*} \{a\}=2, 5,8,11,14,17,20,... \end{equation*} \]

Which can be re-written explicitly:

\[ a_n=2+3n \]

Another type is a geometric sequence. Geometric sequences comprise of the same value being multiplied to the previous term. This is an example:

\[ \begin{cases} a_0=1\\ a_n=a_{n-1}\cdot\frac{1}{3} \end{cases} \]
\[ \{a\}=1,\frac{1}{3},\frac{1}{9},\frac{1}{27},\frac{1}{81},... \]

Which can be re-written explicitly:

\[ a_n=\frac{1}{3^n} \]

A sequence is said to converge if the limit of the series is a real number, \( L \) :

\[ \lim_{n\to\infty}a_n=L \]

Note that if a sequence is made by adding a sequence to a finite number of terms the new sequence will converge if the old one does and will diverge if the old one does.

The squeeze theorem states that if each value in a sequence is between the values of two other sequences and those sequences converge to a number $L$, then the first sequence converges to the same number:

\[ a_n\leq b_n\leq c_n \qquad \lim_{n\to\infty}a_n=\lim_{n\to\infty}c_n=L \]
\[ \lim_{n\to\infty}b_n=L \]

A bounded sequence is a sequence that doesn't get bigger than a specific value (\( M\geq a_n \), bounded above) or smaller than a specific value (\( M\leq a_n \) , bounded below). Note that convergent sequences are bounded, but not all bounded sequences are convergent.

A monotone sequence is one that either only increases or only decreases. The monotone convergent theorem states that if a sequence is monotone and bounded (in the same direction) then it converges.

Example

Use the monotone convergent theorem to find if this sequence converges:

\[ \begin{Bmatrix}\frac{3^n}{n!}\end{Bmatrix} \]

If we write out the first couple of terms, we can see that the first couple of terms get bigger, but after the fourth term, they only get smaller:

\[ 1+3+\frac{9}{2}+\frac{9}{2}+\frac{27}{8}+\frac{81}{40}+\ldots \]

We can see why if we re-write the explicit form of the sequence:

\[ a_{n+1} = \frac{3^{n+1}}{(n+1)!} \]
\[ = \frac{3}{n+1}\cdot \frac{3^n}{n!} \]
\[ = \frac{3}{n+1}\cdot a_n \]

When it is written like this we can see that the next term is smaller than the previous term if \(n\geq2 \) . The sequence is bounded below by 0. The sequence only decreases after the fourth term. Since a finite number of terms added to a sequence doesn't affect the convergence of the sequence we can say that the sequence converges due to the monotone convergent theorem.

Practice

Find an explicit formula for each of these sequences:

  1. \(5,9,14,18,22,26,... \)
  2. \(2,6,18,54,162,... \)
  3. Find the limit of the sequence: \( a_n=\frac{n^2}{2^n} \)

For the following functions, determine whether or not they are bounded and whether or not they are monotone.

  1. \( \sqrt[n]{n}\)
  2. \( \cos(n) \)

Solutions:

  1. \( 5+4n\) starting at \( n=0 \)
  2. \( 2\cdot3^n \) starting at \( n=0 \)
  3. 0
  4. bounded, decreasing monotone
  5. bounded, not monotone

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