Question

Consider the following series: [infinity] 1/n⁶, n = 1. Use the sum of the first 10 terms to estimate the sum of the given series. (Round the answer to six decimal places.)

Answer 1

To estimate the sum of the given series [infinity] 1/n⁶, n = 1 using the sum of the first 10 terms, substitute values of n from 1 to 10 into the formula 1/n⁶, calculate the sum of the first 10 terms, and round the final answer to six decimal places.

Step-by-step explanation:

To estimate the sum of the given series [infinity] 1/n⁶, n = 1 using the sum of the first 10 terms, follow these steps:

1. Substitute the values of n from 1 to 10 into the formula 1/n⁶.
2. Calculate the sum of the first 10 terms by adding up the values obtained in step 1.
3. Round the final answer to six decimal places.

For example:

1/1⁶ + 1/2⁶ + 1/3⁶ + 1/4⁶ + 1/5⁶ + 1/6⁶ + 1/7⁶ + 1/8⁶ + 1/9⁶ + 1/10⁶ = 1 + 0.015625 + 0.00002143347 + 0.00000001526 + 0.00000000006 + 0.0000000000013 + 0.000000000000031 + 0.00000000000000026 + 0.0000000000000000016 + 0.00000000000000000001 = 1.01564644978

Rounding this answer to six decimal places, we get 1.015646.