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Chapter 4: Problem 4

Using sigma notation, write the following expressions as infinite series. $$ \sin 1+\sin 1 / 2+\sin 1 / 3+\sin 1 / 4+\cdots $$

Short Answer

Expert verified
\( \sum_{n=1}^{\infty} \sin \left( \frac{1}{n} \right) \)

Step by step solution

01

Identify the General Term

In the expression \( \sin 1 + \sin \frac{1}{2} + \sin \frac{1}{3} + \sin \frac{1}{4} + \cdots \), observe the pattern of the sequence. Each term can be represented as \( \sin \frac{1}{n} \) where \( n = 1, 2, 3, \ldots \). This pattern helps us write the general term.
02

Setup the Sigma Notation

The sigma notation for an infinite series represents the sum of terms from \( n = 1 \) to infinity. Using the general term identified in Step 1, the sigma notation becomes:\[\sum_{n=1}^{\infty} \sin \left( \frac{1}{n} \right)\]This represents the infinite series where each term is \( \sin \left( \frac{1}{n} \right) \).
03

Confirm the Series Setup

Double-check that the identified pattern for the terms is correctly applied in the sigma notation. Ensure that all terms match the pattern \( \sin \left( \frac{1}{n} \right) \) as test cases like \( n=1, 2, 3, \ldots \) yield the correct terms \( \sin 1, \sin \frac{1}{2}, \sin \frac{1}{3}, \ldots \). This confirms that the sigma notation accurately represents the infinite series.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Infinite Series
An infinite series is a sum of an infinite sequence of terms. It can be represented using sigma notation, which condenses the series into a concise mathematical form. Infinite series are fundamental in calculus and analysis because they allow mathematicians to work with sums that would otherwise be unwieldy.
  • An infinite series has no defined endpoint, as it continues to add terms indefinitely.

  • The convergence or divergence of an infinite series is crucial. Convergence means the series approaches a specific number, while divergence means it increases forever or doesn't approach a fixed limit.

  • Recognizing patterns in the terms is key to setting up an infinite series. In our problem, the pattern is based on trigonometric functions.
Understanding these concepts helps us analyze and determine how a series behaves and whether it can be expressed in a simpler form.
Trigonometric Functions
Trigonometric functions like sin, cos, and tan, are functions of angles and are particularly useful in studying cycles and oscillations. The sine function, used in this problem, is one of the fundamental trigonometric functions that maps angles to values between -1 and 1.
  • The sine of an angle is the y-coordinate of a point on the unit circle corresponding to that angle.

  • The formula for a general sine function is \( y = \sin(x) \), where x is an angle in radians.

  • When creating a series with sine, as in our problem, it involves function values at rational angle divisors, such as \( \sin(1), \sin(\frac{1}{2}), \sin(\frac{1}{3}) \), etc.
These trigonometric values embedded in series are often useful in fields like physics and engineering for analyzing wave patterns and harmonic motions.
Summation Notation
Summation notation, often expressed with the Greek letter sigma (\( \Sigma \)), provides a concise way to represent the sum of a sequence of terms. This notation is especially helpful when dealing with patterns or large numbers of terms.
  • The symbol \( \Sigma \) denotes summation, indicating that you sum up the specified terms.

  • In our case, \( \sum_{n=1}^{\infty} \sin(\frac{1}{n}) \) indicates you are summing the sine values for all \( n \) from 1 to infinity.

  • This notation helps simplify the representation of complex series, making them easier to work with and analyze mathematically.
This concise representation is not only useful in mathematics but also in computing and data analysis, where large datasets can often be summed in a similar manner with predictable patterns.

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Most popular questions from this chapter

The following alternating series converge to given multiples of \(\pi .\) Find the value of \(N\) predicted by the remainder estimate such that the Nth partial sum of the series accurately approximates the left-hand side to within the given error. Find the minimum \(N\) for which the error bound holds, and give the desired approximate value in each case. Up to 15 decimals places, \(\pi=3.141592653589793 .\)[T] The Euler transform rewrites \(S=\sum_{n=0}^{\infty}(-1)^{n} b_{n}\) as \(S=\sum_{n=0}^{\infty}(-1)^{n} 2^{-n-1} \sum_{m=0}^{n}\left(\begin{array}{c}n \\ m\end{array}\right) b_{n-m}\). For the alternating harmonic series, it takes the form \(\ln (2)=\sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n}=\sum_{n=1}^{\infty} \frac{1}{n 2^{n}} .\) Compute partial sums of \(\sum_{n=1}^{\infty} \frac{1}{n 2^{n}}\) until they approximate \(\ln (2)\) accurate to within \(0.0001\). How many terms are needed? Compare this answer to the number of terms of the alternating harmonic series are needed to estimate \(\ln (2)\).

The kth term of each of the following series has a factor \(x^{k}\). Find the range of \(x\) for which the ratio test implies that the series converges. $$ \sum_{k=1}^{\infty} \frac{x^{2 k}}{3^{k}} $$

The kth term of each of the following series has a factor \(x^{k}\). Find the range of \(x\) for which the ratio test implies that the series converges. $$ \sum_{k=1}^{\infty} \frac{x^{k}}{k !} $$

In the following exercises, use an appropriate test to determine whether the series converges. $$ a_{k}=1 /\left(\begin{array}{l} 2 k \\ k \end{array}\right) $$

Suppose a computer can sum one million terms per second of the divergent series \(\sum_{n=1}^{N} \frac{1}{n} .\) Use the integral test to approximate how many seconds it will take to add up enough terms for the partial sum to exceed 100 .
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