Question

In Exercises \(55-58,\) the series represents a well-known function. Use a computer algebra system to graph the partial sum \(S_{10}\) and identify the function from the graph. $$ f(x)=\sum_{n=0}^{\infty}(-1)^{n} \frac{x^{2 n+1}}{(2 n+1) !} $$

Short answer

The function represented by the series is the sine function.

Step-by-step solution

Calculate the partial sum

Use a computer algebra system to calculate the partial sum \(S_{10}\) of the series. To do this, replace the upper limit of the sum from \(\infty\) to \(10\). This gives the partial sum \(S_{10} = \sum_{n=0}^{10}(-1)^{n} \frac{x^{2 n+1}}{(2 n+1) !}\).

Graph the partial sum

With the use of the computer algebra system, graph \(S_{10}\) on a set of axes. Observe the shape, intersection points, and general behavior of the graph.

Identify the function

Compare the graph of the partial sum \(S_{10}\) to the plots of well-known functions. The shape of the graph should resemble one of these functions. After analyzing the graph, it can be determined that the graph matches the graph of sine due to its wave-like characteristics. This leads to the conclusion that the original series represents the sine function.