Series
- Page ID
- 219480
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Definition of a Series
Let an be a sequence then we define the nth partial sum of an as
| sn = a1 + a2 + ... + an |
In other words, we define sn by adding up the first n terms of an. We define the series as the limit of the sn that is
|
S = San = a1 + a2 + a3 + ... |
If the limit exists then we say that the series converges. Otherwise, we say that the series diverges.
Example
consider
1 1
an = -
n2 + 2n + 1 n2
Evaluate
\( S = \displaystyle\sum_{n=1}^{\infty} (\frac{1}{n^2 + 2n + 1} - \frac{1}{n^2} ) \)
Solution
We write out the first four terms:
1 1 1 1 1 1 1 1
- + - + - + - + ...
4 1 9 4 16 9 25 16
1 1 1 1 1 1 1 1
= - + - + - + - + + ...
1 4 4 9 9 16 16 25
= -1
Such a series is called a telescoping series.
Geometric Series
We define a geometric series to be a series of the form
Sarn
For example:
3/2 + 3/4 + 3/8 + ...
|
Geometric Series Test \[ \displaystyle\sum_{n=0}^{\infty} ar^n = \frac{a}{1-r} \]and for |r| > 1 the series diverges. |
Proof:
Let
s = a + ar + ar2 + ar3 + ar4 + ...
Then
rs = ar + ar2 + ar3 + ar4 + ...
subtracting the second equation from the first we get
s - rs = a
or
s(1 - r) = a,
a
s =
1 - r
The Limit Test
|
The Limit Test If \(\displaystyle\sum_{n=0}^{\infty} a_n\) converges then \( \lim\limits_{n \to \infty} a_n = 0 \) |
Note: The contrapositive says that if the limit is nonzero, then the series does not converge.
Caution: If the limit goes to zero then the series still may diverge.
Examples
-
\(\displaystyle\sum_{n=0}^{\infty} \frac{n+3}{n+2}\) diverges by the limit test since the limit is 1 not 0. -
\(\displaystyle\sum_{n=1}^{\infty} \frac{1}{n}\)does not converge even though the limit goes to 0. This series is called the harmonic series.
The Harmonic Series
|
Harmonic Series Test The series with terms 1/n diverges. |
Proof: we write
1 1 1 1 1 1 1 1
+ + + + + + + +
1 2 3 4 5 6 7 8
1 1 1 1 1 1 1 1
+ + + + + + + + + ...
9 10 11 12 13 14 15 16
1 1 1 1 1 1 1 1
> + + + + + + + +
1 2 4 4 8 8 8 8
1 1 1 1 1 1 1 1
+ + + + + + + + + ...
16 16 16 16 16 16 16 16
1 1 1 1
= + + + + ...
1 2 2 2
which diverges by the nth term test. Hence the harmonic series diverges.