Consider the following convergent series. Then complete parts (a) through (d).

a. Find an upper bound for the remainder in terms of . Choose the correct answer.
A.
B.
C.
D.
D

Ask by Gough Mann. in the United States

Mar 30,2025

Question

Consider the following convergent series. Then complete parts (a) through (d).

a. Find an upper bound for the remainder in terms of . Choose the correct answer.
A.
B.
C.
D.
D

Ask by Gough Mann. in the United States

Mar 30,2025

UpStudy AI · Accepted Answer

The series given is $$ \sum_{k=1}^{\infty} \frac{1}{2^{k}} $$, which is a geometric series with a common ratio $$ r = \frac{1}{2} $$. The sum of an infinite geometric series can be calculated using the formula $$ S = \frac{a}{1 - r} $$, where $$ a $$ is the first term.
The first term $$ a = \frac{1}{2} $$. Therefore, the sum of the series is:

$$
S = \frac{\frac{1}{2}}{1 - \frac{1}{2}} = 1
$$

The remainder $$ R_n $$ after $$ n $$ terms of the series can be expressed as:

$$
R_n = S - S_n
$$

where $$ S_n $$ is the sum of the first $$ n $$ terms. The sum of the first $$ n $$ terms of a geometric series is given by:

$$
S_n = \frac{a(1 - r^n)}{1 - r} = \frac{\frac{1}{2}(1 - (\frac{1}{2})^n)}{1 - \frac{1}{2}} = 1 - \frac{1}{2^n}
$$

Thus, the remainder is:

$$
R_n = 1 - (1 - \frac{1}{2^n}) = \frac{1}{2^n}
$$

To find an upper bound for the remainder $$ R_n $$, we can express it in terms of $$ n $$:

$$
R_n = \frac{1}{2^n}
$$

This indicates that the upper bound for the remainder can be expressed as $$ 2^{-(n)} $$. However, we need to check the options provided to find the correct one.
Let's evaluate the options given:

A. $$ 2^{-(n+1)} $$ - This is less than $$ \frac{1}{2^n} $$.

B. $$ \frac{2^{-n}}{\ln 2} $$ - This is not a valid upper bound.

C. $$ 2^{-(n+1)} $$ - This is also less than $$ \frac{1}{2^n} $$.

D. $$ 2^{-n} \ln 2^{n} $$ - This is not a valid upper bound.

The correct upper bound for the remainder in terms of $$ n $$ is $$ 2^{-(n+1)} $$ (option A).

The correct answer for the upper bound of the remainder in terms of $$ n $$ is A. $$ 2^{-(n+1)} $$.

Consider the following convergent series. Then complete parts (a) through (d).

a. Find an upper bound for the remainder in terms of . Choose the correct answer.
A.
B.
C.
D.
D

Solución de inteligencia artificial de Upstudy

Responder

Solución

Beyond the Answer

preguntas relacionadas

Latest Calculus Questions

Consider the following convergent series. Then complete parts (a) through (d). a. Find an upper bound for the remainder in terms of . Choose the correct answer. A. B. C. D. D

Solución de inteligencia artificial de Upstudy

Responder

Solución

Beyond the Answer

preguntas relacionadas

Latest Calculus Questions

Consider the following convergent series. Then complete parts (a) through (d).

a. Find an upper bound for the remainder in terms of . Choose the correct answer.
A.
B.
C.
D.
D