Series of real numbers

Terminology

For any sequence $\left\{a_n\right\}_{n=1}^{\infty}$ of real numbers, the associated series is defined as the ordered formal sum $$\sum_{n=1}^{\infty}a_n=a_1+a_2+a_3\cdots .$$ The sequence of partial sums $\left\{S_k\right\}_{k=1}^{n}$, associated to the series $\sum_{n=1}^{\infty}a_n$, is defined for each $n$ as the sum of the sequence $\left\{a_n\right\}_{n=1}^{\infty}$ from $a_{1}$ to $a_{n}$. That is $$S_n = a_1 + a_2 + a_3 + \cdots + a_n.$$

Simulation

The following simulation plots the partial sums $S_n$ associated to the series $\sum_{n=1}^{\infty}a_n$. It also provides approximate values of $S_n$, for specific values of $n$.

Things to try:

• Drag the slider (n) to explore the values of $S_n$.
• Change the function defining the sequence.
   Example 1: (-1)^(n+1)/n
   Example 2: n*sin(sqrt(n+1))

Worksheet exemplar

The following file contains activities and problems associated with the simulation.

• Series (pdf)
• Series (tex)