Question

Consider the geometric series $f(r)=\sum _{k=0}^{\mathrm{\infty }}{r}^k,$ where $|r|<1$ a. Fill in the following table that shows the value of the series $f(r)$ for various values of $r$ $$\begin{array}{|cccccccccc|}\hline r& -0.9& -0.7& -0.5& -0.2& 0& 0.2& 0.5& 0.7& 0.9\\ f(r)& & & & & & & & & \\ \hline\end{array}$$ b. Graph $f,$ for $|r|<1$ \text { c. Evaluate } \lim _{r \rightarrow 1^{-}} f(r) \text { and } \lim _{r \rightarrow-1^{+}} f(r)

Short answer

Answer: The limits are $\underset{r\to 1^{-}}{lim}f(r)=\mathrm{\infty }$ and $\underset{r\to -1^{+}}{lim}f(r)$ does not exist.

Step-by-step solution

Fill the table for $f(r)$

To find the sum of the geometric series for a given $r$, we can use the formula $$f(r)=\frac{1}{1-r}.$$ To fill the table, we will substitute each value of $r$ into this formula and find the corresponding values of $f(r)$. $$\text{Misplaced hline}$$ $$\text{Misplaced hline}$$

Sketch the graph of $f(r)$

We have filled the table with values of $f(r)$. Now, we will sketch the graph of the function $f(r)=\frac{1}{1-r}$ for $|r|<1$. The function is rational and has a vertical asymptote at $r=1$. It will be continuous for $r$ values between $-1$ and $1$. Since $f(r)$ is a rational function, we can sketch it smoothly by plotting the points from the table.

Compute the limits

We will now compute the limits $\underset{r\to 1^{-}}{lim}f(r)$ and $\underset{r\to -1^{+}}{lim}f(r)$. For $\underset{r\to 1^{-}}{lim}f(r)$, as r approaches 1 from the left ($r<1$), the denominator $1-r$ approaches 0 and the value of $f(r)$ grows without bound. Therefore, $$\underset{r\to 1^{-}}{lim}f(r)=\mathrm{\infty }.$$ For $\underset{r\to -1^{+}}{lim}f(r)$, let's consider the series expansion of $f(r)$ for values of $r$ close to -1. $$f(r)=\sum _{k=0}^{\mathrm{\infty }}{r}^k=1+r+r^2+r^3+\cdots$$ For $r=-1+ϵ$ with $ϵ$ small and positive, the series becomes: $$f(-1+ϵ)=1+(-1+ϵ)+(-1+ϵ{)}^2+(-1+ϵ{)}^3+\cdots$$ The series does not converge as the terms do not approach to zero as k approaches infinity. Thus, the limit $\underset{r\to -1^{+}}{lim}f(r)$ does not exist.

Key concepts

Sum of a Geometric Series

Understanding the sum of a geometric series is fundamental when dealing with sequences in mathematics. Simply put, a geometric series is the sum of the terms of a geometric sequence, where each term after the first is found by multiplying the previous one by a fixed, non-zero number known as the common ratio. For the series to converge, or to have a finite sum, this ratio must have an absolute value less than 1.

The sum of an infinite geometric series can be described by the formula: $S=\frac{a_1}{1-r}$, where $a_1$ is the first term and $r$ is the common ratio. For example, in the series $f(r)=\sum _{k=0}^{\mathrm{\infty }}{r}^k$, the first term is 1 (since $r^0=1$) and the sum is $\frac{1}{1-r}$ provided that $|r|<1$. This formula allows us to quickly determine the total sum of all the terms in the series.

Convergence of Series

In order for a series to be useful in mathematics, especially in applications like analysis and physics, it needs to converge – meaning it adds up to a finite value. The convergence of a series depends heavily on its terms, and for a geometric series, it depends on the common ratio $r$.

For the series $f(r)$, convergence is guaranteed only if the absolute value of $r$ is less than 1 ($|r|<1$). When $r$ meets this condition, it ensures that each successive term becomes smaller and smaller, eventually leading the series towards a finite limit. The step-by-step solution above demonstrates this by showing that as $r$ gets closer to 1 or -1, the value of the series either increases without bound or fails to settle to a particular value, thus diverging.

Rational Functions

A rational function is a function represented by a fraction of two polynomials. In the context of series and particularly geometric series, the sum can be expressed as a rational function when $|r|<1$. Specifically, in our exercise, the sum of the infinite geometric series $f(r)=\frac{1}{1-r}$ is a simple rational function with the numerator being 1 and the denominator being $1-r$.

Rational functions often have interesting features like asymptotes, which are lines that the graph of a function can approach but never touch. In the step-by-step solution, the function $f(r)$ shows a vertical asymptote at $r=1$. This behavior illustrates that the function does not have a value at $r=1$ and emphasizes the significance of the convergence condition for $r$.

Limits of Functions

The limit of a function is a fundamental concept in calculus that describes the behavior of a function as its input approaches a certain value. Limits are essential for understanding the convergence and divergence of series as they allow us to formalize the idea of approaching a value without necessarily reaching it.

In the exercise, we use limits to understand the behavior of the function $f(r)$ as $r$ approaches 1 from the left (denoted by $r\to 1^{-}$) and as $r$ approaches -1 from the right (denoted by $r\to -1^{+}$). The exercise shows that the limit as $r$ approaches 1 is infinity, meaning the function grows without bound. In contrast, as $r$ approaches -1, the series does not approach a fixed number and therefore the limit does not exist. These limit computations are crucial for mapping the behavior of mathematical functions under different conditions.