United States Calculus 2 - Infinite Series

In summary, the conversation discusses determining the number of terms needed to sum in a convergent series to ensure the remainder is less than 10^-5. The exact value of the series is given as ln(128) = 7*sum[k=1,inf] of (-1)^(k+1)/k. The attempt at solving the problem involves using the remainder theorem for alternating series.
  • #1
GreenPrint
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Homework Statement



Determine how many terms of the following convergent series must be summed to be sure that the remainder is less than 10^-5. Although you do not need it, the exact value of the series is given.

ln(128) = 7*sum[k=1,inf] of (-1)^(k+1)/k

Homework Equations





3. The Attempt at a Solution [/b

| ln(128) - 7*sum[k=1,n] of (-1)^(k+1)/k | < 1/10,000
subtracted ln(128) from both sides
|-7*sum[k=1,n] of (-1)^(k+1)/k | < 1/10,000 - ln(128)
simplified the negative on the left hand side and absolute value
7*sum[k=1,n] of 1/k < 1/10,000 - ln(128)
divided through by 7
sum[k=1,n] of 1/k < 1/70,000 - ln(128)/7

I'm unsure were to go from here, thank you for any help you can provide me.
 
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  • #2
GreenPrint said:

Homework Statement



Determine how many terms of the following convergent series must be summed to be sure that the remainder is less than 10^-5. Although you do not need it, the exact value of the series is given.

ln(128) = 7*sum[k=1,inf] of (-1)^(k+1)/k

Homework Equations





3. The Attempt at a Solution [/b

| ln(128) - 7*sum[k=1,n] of (-1)^(k+1)/k | < 1/10,000
subtracted ln(128) from both sides
|-7*sum[k=1,n] of (-1)^(k+1)/k | < 1/10,000 - ln(128)
simplified the negative on the left hand side and absolute value
7*sum[k=1,n] of 1/k < 1/10,000 - ln(128)
divided through by 7
sum[k=1,n] of 1/k < 1/70,000 - ln(128)/7

I'm unsure were to go from here, thank you for any help you can provide me.

There's a remainder theorem for alternating series. Take a look at it.
 

1. What is Calculus 2 - Infinite Series?

Calculus 2 - Infinite Series is a branch of mathematics that deals with the properties and behavior of infinite sequences of numbers. It is an extension of Calculus 1, which focuses on the study of limits, derivatives, and integrals.

2. Why is it important to study Infinite Series in the United States?

Infinite Series is an important mathematical concept that has numerous applications in fields such as physics, engineering, economics, and computer science. It allows us to model and analyze real-world phenomena that involve continuous change, making it an essential tool for problem-solving and decision-making.

3. What topics are covered in Calculus 2 - Infinite Series?

Some of the key topics covered in Calculus 2 - Infinite Series include power series, Taylor and Maclaurin series, convergence and divergence of series, tests for convergence, and applications of series such as approximation and error estimation.

4. Is Calculus 2 - Infinite Series difficult to learn?

Calculus 2 - Infinite Series can be challenging for some students due to its abstract nature and the need for strong algebraic skills. However, with dedication and practice, anyone can grasp the fundamental concepts and excel in the subject.

5. How can I use Calculus 2 - Infinite Series in my daily life?

Infinite Series has many practical applications, such as in calculating compound interest, modeling natural phenomena like population growth or radioactive decay, and optimizing processes in business and technology. It also helps develop critical thinking skills and improves problem-solving abilities, which are valuable in various aspects of life.

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