41–44. {Use of Tech} Remainders and estimates Consider the fol...

Question

41–44. {Use of Tech} Remainders and estimates Consider the following convergent series.

b. Find how many terms are needed to ensure that the remainder is less than 10⁻³.

43. ∑ (k = 1 to ∞) 1 / 3ᵏ

Accepted Answer

For the series, from J equals 1 to infinity, of 2 divided by 5 to the J, determine the minimum number of terms required so that the remainder is less than 2 multiplied by 10 to 3. We have 4 possible answers, being 345, and 6. Now let's denote our remainder term R sent in. As the sun As J equals N + 1 to infinity. Of 2 divided by 5 to the J. Now, what we can do is make use of the integral test remainder estimate, because this is a decreasing positive function. This means if we have the sun, J equals N + 1 to infinity. A sum function F J. This is less than equals to the integral from N to infinity. Of F of X D X. In this case, our F of X will be 2 divided by 5 to the X. Now what we can do is evaluate our integral. We have the integral from in. To infinity Of 2, divided by 5, raised to the X D X. Now, this integral. We can rewrite to be 2 multiplied. From men to infinity Of 5 to the negative XDX. This is of a certain form that we know, the integral of KA X. This is given by 1 divided by a natural log of K, multiplied by K A X plus C. In our case, A is -1, K is 5. So we can now solve our integral. We still have the 2 present, so we have 2 limit as B approaches infinity due to this being an improper interval. This will be multiplied by our integrals results. In this case, the results of our integral. Ends up being -1 divided by natural log of 5. Multiplied by 5 to the nave of X. From N to be. We can now simplify This is 2 limit as B approaches infinity. Of the results of our integral 1 divided by 5 ton. Natural log 5. -1 divided by 5 to the B natural log of 5. Now we can evaluate our limit. Because we have 1 divided by 5 to the infinity, our second term goes to 0. Meaning we have 2 multiplied by 1 divided by 5 to the end natural log of 5. We can just combine these two values to give us 2 divided by 5 to the end natural log of 5. Now this is our remainder term. Now we need to find our exact number of terms. We have our sub N is less than our error bound, 2 multiplied by 10 to the negative 3. So, we have 2 divided by 5 to the end, natural log 5, less than 0.002. Now we can solve. We can take a reciprocal and multiply some values to get 5 to the end is greater than 2 divided by 0.002. Natural like effect. Now we just need to solve for in. We get natural luck. A 5 to the end. It's greater than natural log of 1000. Divided by natural log 5. Which gives us in Naturalo 5 It's greater than Natural Of 1000 divided by natural like 5. This gives us a value of N being greater than 3.996. We can round this up to give our next smallest integer N equals 4. So, we can determine at least 4 terms are needed. That means the answer to our problem is answer B. OK, I hope to help you solve the problem. Thank you for watching. Goodbye.

41–44. {Use of Tech} Remainders and estimates Consider the following convergent series.

b. Find how many terms are needed to ensure that the remainder is less than 10⁻³.

43. ∑ (k = 1 to ∞) 1 / 3ᵏ

Key Concepts

Geometric Series

Remainder (Error) in Infinite Series

Convergence Criteria and Error Bounds

Watch next