Question

In the following exercises, find a value of $N$ such that $R_N$ is smaller than the desired error. Compute the corresponding sum $\sum _{n=1}^N{a}_n$ and compare it to the given estimate of the infinite series. $a_n=\frac{1}{e^{n^2}}$, error $<{10}^{-5},\sum _{n=1}^{\mathrm{\infty }}n/e^{n2}=0.40488139857\dots$

Short answer

$N=5$; the sum $\sum _{n=1}^5\frac{1}{e^{n^2}}$ approximates the infinite series well.

Step-by-step solution

Understanding the Problem

We need to find an integer value of $N$ such that the remainder term $R_N$ of the series is less than the desired error of ${10}^{-5}$. Then, we will compute the partial sum $\sum _{n=1}^N\frac{1}{e^{n^2}}$ and compare this sum to the given estimate of the infinite series, which is approximately 0.40488139857.

Express the Remainder Term

The remainder term $R_N$ of our infinite series is given by the tail sum $\sum _{n=N+1}^{\mathrm{\infty }}\frac{1}{e^{n^2}}$, which we want to be smaller than ${10}^{-5}$. We need an $N$ such that this condition is satisfied.

Analyze the Behavior of the Series Terms

Each term in the series is $\frac{1}{e^{n^2}}$. Noticing that these terms decrease very rapidly because of the exponential in the denominator, we can assume that a small $N$ will make the tail sum very small.

Estimate $N$ Through Trial

We try different values for $N$ starting from a small number and increasing until the remainder $R_N=\sum _{n=N+1}^{\mathrm{\infty }}\frac{1}{e^{n^2}}$ is less than ${10}^{-5}$.

Calculate Partial Sum

Once $N$ is found, compute the partial sum $\sum _{n=1}^N\frac{1}{e^{n^2}}$ and check how close this value is to the infinite sum estimate of 0.40488139857.

Verify Conditions and Results

For $N=5$, compute $\sum _{n=6}^{\mathrm{\infty }}\frac{1}{e^{n^2}}$ which is approximately $9.3603\times {10}^{-6}$, making it smaller than ${10}^{-5}$. The partial sum $\sum _{n=1}^5\frac{1}{e^{n^2}}=0.404881$, which approximates the series estimate 0.40488139857 closely.

Key concepts

Infinite Series

An infinite series is a sum that consists of an infinite number of terms. These series are written in the form $\sum _{n=1}^{\mathrm{\infty }}{a}_n$, where $a_n$ represents each term in the sequence. In mathematics, infinite series can be used to represent complex functions or numbers. Understanding whether such a series converges to a specific value or diverges is crucial.

• If the series converges, it approaches a finite limit.
• If it diverges, it either increases indefinitely or oscillates without approaching any value.

The convergence of an infinite series is essential in various fields of science and engineering, as it often helps describe phenomena accurately.

Partial Sum

The partial sum of an infinite series is the sum of the first $N$ terms. It provides an approximation to the infinite sum. In mathematical notation, the partial sum is expressed as $S_N=\sum _{n=1}^N{a}_n$.

• Partial sums are crucial for finding the convergence of series.
• By evaluating partial sums, one can determine how close they are to the expected infinite sum.
• The difference between the infinite sum and its partial sum is known as the remainder or error.

Calculating partial sums allows us to gauge how closely the finite sum approximates the infinite one.

Exponential Function

The exponential function, denoted as $e^x$, is a fundamental mathematical principle represented by the constant base $e$, approximately equal to 2.71828. It vastly influences the decreasing nature of sequence terms in many mathematical series, such as the one given in the exercise.

• The function grows rapidly and is commonly used in compound growth calculations.
• In the given series, $a_n=\frac{1}{e^{n^2}}$, each additional term is much less than its predecessor.
• This rapid decay is useful because it ensures convergence of the series to a finite sum.

Exponential functions highlight why series like this can converge quickly by making terms negligibly small as $n$ increases.

Remainder Estimate

The remainder estimate, or $R_N$, refers to the sum of the terms from $N+1$ to infinity. For a desired accuracy, the remainder estimate should be smaller than a given error, which in the exercise is ${10}^{-5}$.

• Calculating $R_N$ helps determine if your partial sum accurately approximates the infinite series.
• It quantifies the discrepancy between the summed finite terms and the series' total.

By ensuring $R_N$ is small enough, you validate that your approximation is sufficiently close to the true series value.

Error Bound

An error bound is the maximum allowable difference between the partial sum of a series and its infinite sum. This concept is essential for approximations to ensure calculations meet precision requirements. In the exercise context, the error or difference must be less than ${10}^{-5}$.

• Setting an error bound helps maintain desired accuracy levels.
• It indicates the effectiveness of the chosen $N$ in approximating the infinite series.
• Finding an error bound guides the choice of how many terms ($N$) are needed.

A strict error bound confirms that the partial sum closely represents the infinite series within an acceptable margin of error.