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Sequences - Calculus Tutorial

A sequence \(a_n\) corresponds to infinite array or list of number of the form

\[a_1, a_2, a_3, ....\]

where \(a_1, a_2, a_3, ...\) are real numbers. For example, the sequence

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

is represented by the list

\[1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, ....\]

because those are the values the expression \(a_n = \frac{1}{n}\) takes when \(n\) takes the values 1, 2, 3, ...etc.

Convergence of sequences

One concept that is typically hard to grasp is the convergence of a sequence. The idea is very trivial though: A sequence \(a_m\) converges to a value \(a\) if the values of the sequence get closer and closer to \(a\) (in fact they get as close as we want) as \(n\) approaches to infinity.

For example: The sequence \(a_n = 1/n\) is such that

\[a_n = \frac{1}{n} \to 0\]

because the value of \(1/n\) becomes "as close to zero as we want" as \(n\) approaches to infinity.

Formal defintion of convergence:

The sequence \(a_n \to a\) as \(n \to \infty\), or otherwise said \(\lim_{n \to \infty}{a_n} = a\) if

• For all \(\varepsilon >0\), there exist \(n_0\) such that \(n \geq n_0 \,\,\, \Rightarrow \,\,\, |a_n - a|< \varepsilon \)

This is saying that no matter how close you want the sequence from \(a\), there is always a point in the sequence such that all the points further than that, are close enough to \(a\). In other words the convergence of a sequence doesn't state that some number of the sequence get close enough to the limit \(a\), but instead, it indicates that if we go far enough into the sequence, all the values of if will be close enough.

Algebra of Limits

Operating with limits is not as complicated once we know some them. In fact, there are simple rules that allow to compute more complicated limits based on simpler ones. These rules are shown below:

If \(\displaystyle\lim_{n \to \infty}{a_n} = a\) and \(\displaystyle\lim_{n \to \infty}{b_n} = b\) then we have:

(1) \(\displaystyle\lim_{n \to \infty}(a_n + b_n) = \displaystyle\lim_{n \to \infty}{a_n}+\displaystyle\lim_{n \to \infty}{b_n} = a + b \)

(2) \(\displaystyle\lim_{n \to \infty}{a_n b_n} = \displaystyle\lim_{n \to \infty}{a_n}\times\displaystyle\lim_{n \to \infty}{b_n} = a b \)

(3) \(\displaystyle\lim_{n \to \infty}\frac{a_n}{b_n} = \frac{\displaystyle\lim_{n \to \infty}{a_n}}{\displaystyle\lim_{n \to \infty}{b_n}} = \frac{a}{b} \)

(where property (3) holds as long as \(b \ne 0 \).)

Example: The limit

\[\lim_{n \to \infty}\frac{n^2}{n^2 + 1}\]

is computed by first multiplying both numerator and denominator by \(\frac{1}{n^2}\), which means

\[\lim_{n\to\infty}\frac{n^2}{n^2 + 1} = \lim_{n\to\infty} \frac{1}{1 + \frac{1}{n^2}}= \frac{1}{1} = 1\]

because \( \displaystyle\lim_{n\to\infty} \frac{1}{n^2} = 0\).

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