All Flashcards
What does the Alternating Series Error Bound Theorem state?
For a convergent alternating series, the error in approximating the sum by the nth partial sum is no greater than the absolute value of the (n+1)th term.
How is the Alternating Series Error Bound Theorem applied?
It's used to estimate the accuracy of approximating the sum of a convergent alternating series with a finite number of terms.
What conditions must be met for the Alternating Series Error Bound Theorem to apply?
The series must be alternating, and the absolute value of the terms must be decreasing and approaching zero.
What is the significance of the Alternating Series Error Bound Theorem in numerical analysis?
It provides a practical way to determine the accuracy of numerical approximations of alternating series.
How does the Alternating Series Error Bound Theorem relate to the concept of convergence?
It relies on the convergence of the alternating series to provide a meaningful error bound.
What is the practical implication of the Alternating Series Error Bound Theorem?
It allows us to determine how many terms are needed to achieve a desired level of accuracy in approximating the sum of an alternating series.
State the Alternating Series Error Bound Theorem in mathematical notation.
If ( |a_{n+1}| \geq |a_{n+2}| ) and ( \lim_{n \to \infty} a_n = 0 ), then ( |s - s_n| \leq |a_{n+1}| ).
How can you use the Alternating Series Error Bound Theorem to find the range of the true sum ( s )?
By setting up the inequality ( s_n - |a_{n+1}| \leq s \leq s_n + |a_{n+1}| ).
What is the role of the condition ( \lim_{n \to \infty} a_n = 0 ) in the Alternating Series Error Bound Theorem?
It ensures that the terms are decreasing and approaching zero, which is necessary for the series to converge and for the error bound to be valid.
How does the Alternating Series Error Bound Theorem help in approximating infinite sums?
It provides a way to quantify the error when approximating an infinite sum with a finite partial sum, allowing for controlled accuracy.
What is the formula for the Alternating Series Error Bound?
In the error bound formula, what does ( s ) represent?
The true sum of the infinite alternating series.
In the error bound formula, what does ( s_{i-1} ) represent?
The partial sum of the first ( i-1 ) terms of the series.
What does ( a_i ) represent in the Alternating Series Error Bound formula?
The absolute value of the ( i )-th term of the series (the first omitted term).
What does the inequality ( |s - s_{i-1}| leq a_i ) imply?
The absolute difference between the true sum ( s ) and the partial sum ( s_{i-1} ) is less than or equal to ( a_i ).
How can you express the range in which the true sum ( s ) lies, given the error bound?
By rewriting the inequality as ( s_{i-1} - a_i leq s leq s_{i-1} + a_i ).
What is the formula to calculate the nth term of the alternating series?
If the alternating series is given by , then the nth term is .
How do you calculate the partial sum ( s_n ) of an alternating series?
Sum the first ( n ) terms of the series: ( s_n = \sum_{i=1}^n (-1)^i a_i ).
Write the general form of an alternating series.
or , where ( a_n > 0 ) for all ( n ).
How can you determine the error bound for the sum of the first ( n ) terms?
The error bound is given by the absolute value of the (n+1)-th term, i.e., ( |a_{n+1}| ).
How do you find the error bound for the sum of the first 5 terms of an alternating series?
Identify the 6th term, , and calculate its absolute value. This value is the error bound.
How do you estimate the true value of an alternating series using the error bound?
Calculate the partial sum . Then, is between and .
Given an alternating series, how do you determine if using the first 4 terms gives an accurate estimation?
Calculate the error bound . A smaller indicates a more accurate estimation.
How to calculate the error bound of ( \sum_{n=1}^{10} \frac{(-1)^n}{n^2} )?
Find the absolute value of the 11th term: ( |a_{11}| = |\frac{(-1)^{11}}{11^2}| = \frac{1}{121} ).
How to find the interval containing the true sum of ( \sum_{n=1}^{\infty} \frac{(-1)^n}{n} ) using the first 5 terms?
- Calculate ( s_5 = \sum_{n=1}^{5} \frac{(-1)^n}{n} ). 2. Find the error bound ( a_6 = \frac{1}{6} ). 3. The interval is ( [s_5 - a_6, s_5 + a_6] ).
How to determine the number of terms needed to estimate the sum with an error less than 0.01?
Find ( n ) such that ( |a_{n+1}| < 0.01 ).
Given ( \sum_{n=1}^{\infty} \frac{(-1)^n}{n^3} ), how to find the error bound after summing the first 3 terms?
The error bound is the absolute value of the 4th term: ( |a_4| = |\frac{(-1)^4}{4^3}| = \frac{1}{64} ).
How do you decide which of two partial sums gives a more accurate estimation?
Calculate the error bound for each partial sum. The partial sum with the smaller error bound is more accurate.
How do you set up the inequality to find the range of the true sum ( s ) given ( s_n ) and ( a_{n+1} )?
Set up the inequality ( s_n - a_{n+1} \leq s \leq s_n + a_{n+1} ).
How do you use the error bound to improve the estimation of the sum of an alternating series?
By narrowing the interval in which the true sum lies, i.e., ( s \in [s_n - a_{n+1}, s_n + a_{n+1}] ).