Question

Given any infinite series $\sum a_k$ let $N(r)$ be the number of terms of the series that must be summed to guarantee that the remainder is less than ${10}^{-r}$ in magnitude, where $r$ is a positive integer. a. Graph the function $N(r)$ for the three alternating $p$ -series $\sum _{k=1}^{\mathrm{\infty }}\frac{(-1{)}^{k+1}}{k^p},$ for $p=1,2,$ and $3.$ Compare the three graphs and discuss what they mean about the rates of convergence of the three series. b. Carry out the procedure of part (a) for the series $\sum _{k=1}^{\mathrm{\infty }}\frac{(-1{)}^{k+1}}{k!}$ and compare the rates of convergence of all four series.

Short answer

The remainder expression for the alternating p-series is given by $R_n\le \frac{1}{(n+1{)}^p}$ where p takes values 1, 2, and 3. The N(r) function, which represents the number of terms required to achieve a given threshold, can be found by solving the inequality $R_n\le {10}^{-r}$ for n. By comparing the N(r) graphs for different p-series, we can observe that the convergence rates of the series increase as the value of p increases. For a new series $\sum _{k=1}^{\mathrm{\infty }}\frac{(-1{)}^{k+1}}{k!}$, the remainder expression is $R_n\le \frac{1}{(n+1)!}$. By finding the N(r) function for this new series and comparing it to the previous p-series, we can observe that it converges faster than any of the given p-series because its denominator increases factorially, leading to a faster rate of convergence.

Step-by-step solution

Find the expression for the remainder of the series.

We have the remainder $R_n\le |a_{n+1}|$ after summing up the first n terms for each series. Thus, we have $R_n\le \frac{1}{(n+1{)}^p}$ where p takes values 1, 2, and 3 for each given series. Step 2: N(r) function for each p-series

Find the N(r) function and graph.

We are looking for the smallest n such that $R_n\le {10}^{-r}$, which is our N(r). $R_n\le \frac{1}{(n+1{)}^p}\le {10}^{-r}$. Solving this inequality for n, we get $n\ge ({10}^r{)}^{\frac{1}{p}}-1$. We can find the smallest integer value that satisfies this inequality and graph N(r) as a function of r for p=1,2,3. Step 3: Comparing the graphs and discussing the convergence rates

Compare the graphs.

After graphing N(r) for each p-series, we can observe how the convergence rates change as the value of p increases. As p increases, the number of terms necessary to achieve a given threshold decreases, indicating that the series converges faster as p increases. Part b: For the new series $\sum _{k=1}^{\mathrm{\infty }}\frac{(-1{)}^{k+1}}{k!}$, we will follow Steps 1 and 2 to find the N(r) function and then compare it to the N(r) of the previous three series. Step 4: Remainder expression for the new series

Find the expression for the remainder of the new series.

We have the remainder $R_n\le |a_{n+1}|$ after summing up the first n terms. Thus, we have $R_n\le \frac{1}{(n+1)!}$. Step 5: N(r) function for the new series

Find the N(r) function and graph.

We are looking for the smallest n such that $R_n\le {10}^{-r}$, which is our N(r). $\frac{1}{(n+1)!}\le {10}^{-r}$. Solving this inequality for n, we find the smallest integer value that satisfies it. Then, graph the new N(r) function as a function of r. Step 6: Comparing the rates of convergence of all four series

Compare the convergence rates.

By comparing the N(r) function for each of the four series, we can notice how quickly the new series converges compared to the three p-series. As expected, the new series converges faster than any of the given p-series because as the integer gets larger, the denominator increases factorially, leading to a faster rate of convergence.

Key concepts

Exploring Alternating Series

Alternating series are special types of infinite series where the terms alternate in sign. This often results in a sequence like:

• The first term is positive.
• The second term is negative.
• And so on.

These series have unique properties, especially when it comes to convergence. A common example is the alternating harmonic series: $$\sum _{k=1}^{\mathrm{\infty }}\frac{(-1{)}^{k+1}}{k}$$ The main advantage of alternating series is that they tend to be easier to determine convergence for, thanks to the Alternating Series Test. This test shows that if the absolute value of the terms decreases steadily to zero, then the series converges. Alternating series frequently appear in calculus and analysis, making them fundamental for understanding more complex mathematical concepts.

Understanding Convergence Rates

Convergence rates help us understand how quickly a series approaches its limit. In the context of alternating series, there's a direct relation between the convergence rate and the exponent value, such as in a series like $\sum _{k=1}^{\mathrm{\infty }}\frac{(-1{)}^{k+1}}{k^p}$.
If the exponent $p$ is higher, the series tends to converge faster. This is because the terms become smaller more quickly, and fewer terms are needed to get close to the sum.
The function $N(r)$ represents the number of terms needed to keep the remainder under a specific threshold. Graphing $N(r)$ for different values of $p$ (like 1, 2, and 3) displays how increasing $p$ reduces $N(r)$.

• Higher $p$ – faster convergence.
• Lower $p$ – slower convergence.

This illustrates a fundamental property of series and helps in comparing different series.

Factorial Series and Their Impact

Factorial series, such as $\sum _{k=1}^{\mathrm{\infty }}\frac{(-1{)}^{k+1}}{k!}$, converge exceptionally fast. The reason lies in the nature of the factorial $!$ in the denominator.
A factorial grows very rapidly. For example:

• $5!=5\times 4\times 3\times 2\times 1=120$

This rapid growth means the terms in the factorial series become tiny very quickly. The remainder of the series, therefore, converges to zero much faster than in standard $p$-series.
The graph of $N(r)$ for a factorial series will typically show a much smaller $N(r)$ compared to any $p$-series, indicating fewer terms are needed. This property of factorials makes such series important in various fields, like physics and engineering, where efficient calculations are crucial.

Graphing Functions for Series Analysis

Graphing functions such as $N(r)$ helps visualize and compare the convergence of different series. These functions plot $r$ (which controls the precision of the remainder) against the number of terms needed, $N(r)$.
By graphing $N(r)$ for each series, we can quickly see which series converges more rapidly. The graphs for different $p$-series:

• The graph flattens more as $p$ increases.
• Lower graphs represent slower convergence.

For factorial series, the graph would drop steeply, showing a much lower $N(r)$.
This type of analysis is especially helpful in science and mathematics for visual learners. It underscores the importance of being able to interpret mathematical data graphically.