8.2 Series

Mathematica script by Chris Parrish,
   cparrish@sewanee.edu

Sources and references for some of these problems include
    James Stewart, "Calculus: Concepts and Contexts," Second Edition, Brooks/Cole, 2001
    Deborah Hughes-Hallett, Andrew M. Gleason, et. al., "Calculus," Second Edition, John Wiley & Sons, 1998
    Robert Fraga, ed., "Calculus Problems for a New Century," The Mathematical Association of America, 1993
    Selwyn Hollis, "CalcLabs with
Mathematica" for Stewart's "Single Variable Calculus, Concepts and Contexts, Second Edition," Brooks/Cole, 2001
    
Credits: In these examples we follow a programming style used by
Selwyn Hollis in "CalcLabs with Mathematica," Chapter 7, "Sequences and Series."

Partial Sums

Partial Sums of Infinite Series

Reference: In these examples we follow a programming style used by Selwyn Hollis in "CalcLabs with Mathematica," Chapter 7, "Sequences and Series."

The Mathematica command "Table" can be used to generate the partial sums of infinite series.

In[98]:=

<<Graphics`Colors`

In[99]:=

Clear[a, k, n, seq, partialSums]  a[k_] := 1/k^2 seq = Table[a[k], {k, 20}] pa ...                                                                                                 k

Out[101]=

{1, 1/4, 1/9, 1/16, 1/25, 1/36, 1/49, 1/64, 1/81, 1/100, 1/121, 1/144, 1/169, 1/196, 1/225, 1/256, 1/289, 1/324, 1/361, 1/400}

Out[102]=

{1, 5/4, 49/36, 205/144, 5269/3600, 5369/3600, 266681/176400, 1077749/705600, 9778141/6350400, ... 1143261/150117385017600, 86364397717734821/54192375991353600, 17299975731542641/10838475198270720}

[Graphics:HTMLFiles/8.2_series_5.gif]

This sequence converges, and Mathematica reports the value of the infinite sum.

In[104]:=

Underoverscript[∑, k = 1, arg3] 1/k^2

Out[104]=

π^2/6

Some Important Series

Geometric Series

The geometric series is one of the simplest series. It arises often in applications and it can be used to determine the convergence of other more complicated series.

In[105]:=

Clear[r, n] ;  n = 10 ;  seq = Table[r^k, {k, 0, n}]  Underoverscript[∑, k = 0, arg3] r^k

Out[107]=

{1, r, r^2, r^3, r^4, r^5, r^6, r^7, r^8, r^9, r^10}

Out[108]=

1 + r + r^2 + r^3 + r^4 + r^5 + r^6 + r^7 + r^8 + r^9 + r^10

The geometric series converges iff -1 < r < 1.

In[109]:=

Underoverscript[∑, k = 0, arg3] r^k

Out[109]=

1/(1 - r)

Notice that Mathematica assumes that r is in the proper range for convergence of the geometric series.
Here is an illustration of that convergence for a specific value of r.

In[110]:=

Clear[r, k, n, seq, partialSums]  r = 1/2 ;  seq = Table[r^k, {k, 0, 20}] part ...                                                                                             k = 1

Out[112]=

{1, 1/2, 1/4, 1/8, 1/16, 1/32, 1/64, 1/128, 1/256, 1/512, 1/1024, 1/2048, 1/4096, 1/8192, 1/16384, 1/32768, 1/65536, 1/131072, 1/262144, 1/524288, 1/1048576}

Out[113]=

{1, 3/2, 7/4, 15/8, 31/16, 63/32, 127/64, 255/128, 511/256, 1023/512, 2047/1024, 4095/2048, 81 ... 7/16384, 65535/32768, 131071/65536, 262143/131072, 524287/262144, 1048575/524288, 2097151/1048576}

[Graphics:HTMLFiles/8.2_series_16.gif]

p-Series

The p-series is another important series. It can be used to determine the convergence of other more complicated series.

In[115]:=

Clear[p, m, n] ;  p = 2 ; m = 10 ;  seq = Table[1/n^p, {n, m}]  Underoverscript[∑, n = 1, arg3] 1/n^p

Out[118]=

{1, 1/4, 1/9, 1/16, 1/25, 1/36, 1/49, 1/64, 1/81, 1/100}

Out[119]=

1968329/1270080

The p-series converges iff p > 1.

In[120]:=

Underoverscript[∑, n = 1, arg3] 1/n^p

Out[120]=

π^2/6

The following illustration suggests that convergence.

In[121]:=

Clear[p, m, n, seq, partialSums]  p = 2 ;  seq = Table[1/n^p, {n, 20}] partial ...                                                                                                 n

Out[123]=

{1, 1/4, 1/9, 1/16, 1/25, 1/36, 1/49, 1/64, 1/81, 1/100, 1/121, 1/144, 1/169, 1/196, 1/225, 1/256, 1/289, 1/324, 1/361, 1/400}

Out[124]=

{1, 5/4, 49/36, 205/144, 5269/3600, 5369/3600, 266681/176400, 1077749/705600, 9778141/6350400, ... 1143261/150117385017600, 86364397717734821/54192375991353600, 17299975731542641/10838475198270720}

[Graphics:HTMLFiles/8.2_series_25.gif]

Harmonic Series

The harmonic series is the most famous divergent series.

In[126]:=

Clear[k, n] ;  n = 10 ;  Underoverscript[∑, k = 1, arg3] 1/k

Out[128]=

7381/2520

Let's plot some partial sums of the harmonic series.

In[129]:=

Clear[harmonicPartialSum, partialSums, n] ;  harmonicPartialSum[n_] := harmonicPS[n] & ... p;       , AxesLabel {"n", None}}], ]}], ;}]

Out[131]=

{1, 3/2, 11/6, 25/12, 137/60, 49/20, 363/140, 761/280, 7129/2520, 7381/2520, 83711/27720, 8602 ... 60360, 2436559/720720, 42142223/12252240, 14274301/4084080, 275295799/77597520, 55835135/15519504}

[Graphics:HTMLFiles/8.2_series_30.gif]


Created by Mathematica  (May 5, 2004)