6 Sequent Sum

A sequence is an ordered list of terms. It can be finite or infinite. Some famous sequences include:

Sequence	Description
\(1, 4, 9, 16, 25, \dots\)	The square numbers. The sequence of \(n^2\) for \(n \geq 1\).
\(1, 3, 6, 10, 15, 21, \dots\)	The triangular numbers. A sequence representing sums of natural numbers, with the formula \(T_n=\frac{n(n+1)}{2}\).
\(2, 3, 5, 7, 11, 13, 17, \dots\)	The prime numbers, which are natural numbers which are not the product of two smaller natural numbers.

6.1 Sequences with `arange`

The NumPy library has a useful function called arange for generating sequences as an array over a particular range.

import numpy as np 
print( np.arange(10) )

[0 1 2 3 4 5 6 7 8 9]

By default, arange starts at 0 and the stopping number (the 10 above) is excluded. We can pass arange more information about how we want to build the seqeuence in the form: \[\text{np.arange(start, stop, step)}\] which specifies the start, the (excluded) stop, and the step size. The numbers 1 through 10 are generated with:

np.arange(1, 11, 1)

array([ 1,  2,  3,  4,  5,  6,  7,  8,  9, 10])

The even numbers from 20 through 30:

np.arange(20, 31, 2)

array([20, 22, 24, 26, 28, 30])

Here are examples of sequences with NumPy:

Sequence	Python
\(1,2,3,4,5\)	`np.arange(1,6)`
\(1,3,5,7,\dots,99\)	`np.arange(1,100,2)`
\(10,20,30,\dots,100\)	`np.arange(10,101,10)`
\(10,9,8,7,\dots,1\)	`np.arange(10,0,-1)`

6.2 Sequences with `linspace`

Another useful NumPy function is linspace, which generates a sequence with linearly spaced values. The syntax for linspace is: \[\text{np.linspace(start, stop, num)}\] where \(\text{num}\) is the number of elements in the sequence (as opposed to the step size). The line of code:

np.linspace(0,10,50)

array([ 0.        ,  0.20408163,  0.40816327,  0.6122449 ,  0.81632653,
        1.02040816,  1.2244898 ,  1.42857143,  1.63265306,  1.83673469,
        2.04081633,  2.24489796,  2.44897959,  2.65306122,  2.85714286,
        3.06122449,  3.26530612,  3.46938776,  3.67346939,  3.87755102,
        4.08163265,  4.28571429,  4.48979592,  4.69387755,  4.89795918,
        5.10204082,  5.30612245,  5.51020408,  5.71428571,  5.91836735,
        6.12244898,  6.32653061,  6.53061224,  6.73469388,  6.93877551,
        7.14285714,  7.34693878,  7.55102041,  7.75510204,  7.95918367,
        8.16326531,  8.36734694,  8.57142857,  8.7755102 ,  8.97959184,
        9.18367347,  9.3877551 ,  9.59183673,  9.79591837, 10.        ])

creates a numpy array that starts at 0, ends at 10, and has a total of 50 equally spaced elements in the array. The linspace function will be particularly useful for graphing functions in Section 7.1.

6.3 Operations on Arrays

Given a sequence as a NumPy array, we can perform math operations which apply to every element in the sequence.

x = np.array([1, 2, 3, 4, 5])

x + 4      # [5, 6, 7, 8, 9, 10]
x - 1      # [0, 1, 2, 3, 4]
5 * x      # [5, 10, 15, 20, 25]
x / 4      # [0.25, 0.5, 0.75, 1., 1.25] 
x**2       # [1, 4, 9, 16, 25]

Given two sequences, it is important to know that the standard operations are performed element-wise, meaning that the operation is performed between matching pairs of elements.

x = np.array([1, 2, 3])
y = np.array([4, 5, 6])

x + y      # [5, 7, 9]
x - y      # [-3, -3, -3]
x * y      # [4, 10, 18]
x / y      # [0.25, 0.4, 0.5]

6.4 Sums

A famous math problem asks for the sum of the numbers from 1 to 100, \[\sum_{n=1}^{100}n\] which we can find quickly with:

np.sum( np.arange(1,101,1) )

Given a sum with a more complex expression, we can find that sum by initializing the iteration variable and then writing the expression. For example,
\[\sum_{k=1}^{20}k^2+3k+2\] can be found with:

k = np.arange(1,21,1)        # Creates sequence k = 1 through 20
np.sum( k**2 + 3*k + 2 )     # Sums k^2+3k+2 for each element in k

Here are some more examples:

Series	Python
\[\sum_{n=0}^{10}\frac{1}{2^{n}}\]	`n = np.arange(11)` `np.sum( 1/2**n )`
\[\sum_{m=5}^{15}\frac{m+m^3}{3+m}\]	`m = np.arange(5,16)` `np.sum( (m+m^3) / (3+m) )`
\[\sum_{x=1}^{20}3x-5\]	`x = np.arange(1,21)` `np.sum( 3*x-5 )`

Exercises

Generate two NumPy arrays, each representing one million rolls of a die. Create a histogram of their sum. Be sure to specify the correct number of bins and label the plot.
Create each of the following arithmetic sequences using a single line of code for each:
- \(1,2,3,4,5,6\)
- \(2,4,6,8,10\)
- \(5,10,...,40,45\)
- \(32,29,...,-7,-10\)
- Odd numbers from 77 to 101
- Even numbers from 60 down to 40
Write the most concise NumPy expressions you can to evaluate the summations of the following sequences:
- Integers from 1 to 100
- Even numbers from 2 to 100
- \(-18, -12, -6, 0, 6, 12, 18, 24\)
- \(1.5, 4, 6.5, 9, ..., 49\)
- \(0,\frac{\pi}{6},\frac{\pi}{3},\frac{\pi}{2},\frac{2\pi}{3},...,2\pi\)
- Odd numbers greater than 100 and less than 1000
- All the prime numbers less than 100
Find the value of the following sums using NumPy:

\[\sum_{k=1}^{5}k^2-3\]	\[\sum_{n=1}^{5}(4-n^2)\]	\[\sum_{m=1}^{10}\left(4m^2-2\right)\]
\[\frac{7}{4}+\frac{9}{5}+\frac{11}{6}+\dots+\frac{107}{54}\]	\[\sum_{h=4}^{9}h(h+5)\]	\[\sum_{m=4}^{9}\frac{m^2+1}{m}\]
\[\sum_{x=-4}^{4}x^2+4x+5\]	\[\sum_{n=1}^{100}\frac{1}{n}\]	\[\frac{10}{1}+\frac{20}{2}+\frac{30}{3}+\dots+\frac{100}{10}\]

6.1 Sequences with arange

6.2 Sequences with linspace

6.3 Operations on Arrays

6.4 Sums

Exercises

6.1 Sequences with `arange`

6.2 Sequences with `linspace`