Sequences

Python’s built in for loops are a great tool at solving these kinds of problems. Let’s take a look.

Evaluating series

This can be solved with a simple for loop.

For example $\sum_{r=1}^{20}{\left(r+1\right)}^3$ can be solved with the following code:

s_sum=0

for r in range(1,20+1):
    s_sum += (r+1)**3

print("The sum of series is", s_sum)

Try it out

```

Writing sequences

Sequences can be written in sympy with the sequence(x, (x, a, b)) function. Here x represents a mathematical symbol and this is declared with x = Symbol('x'), whilst a and b define the [a, b] interval which the terms of the series are in.

Here is an example to write the sequence n^2 for n bigger than 0:

from sympy import *
from sympy.abc import n

s = SeqFormula(n**2, (n, 0, oo)) # oo is infinity

The following functions are useful when using sequences in sympy:

• s.coeff(i) returns the ith term of the sequence s
• s.formula shows the formula used in the sequence s

Try it out

Divergence test

Sympy is also a great tool to easily check if a series is convergent or divergent.

Let’s check if $\sum_{n=1}^{\infty{}}\frac{n}{n^2+1}$ and $\sum_{n=0}^{\infty{}}\frac{1}{\left(2n+1\right)!}$ are divergent or absolutely convergent.

from sympy import Sum, oo
from sympy.abc import n

Sum((-1)**n, (n, 1, oo)).is_absolutely_convergent() # returns false
Sum(1/factorial(2*n+1), (n, 1, oo)).is_absolutely_convergent() # returns true