Final answer:
To approximate the sum of the series ∑[n=1 to 10] 1 / (9 + n^5), we can use the sum of the first 10 terms and calculate their sum. The approximate sum is 0.14494 (rounded to five decimal places). However, we don't have enough information to estimate the error using the Remainder Estimation Theorem.
Step-by-step explanation:
To approximate the sum of the series ∑[n=1 to 10] 1 / (9 + n^5), we can use the sum of the first 10 terms. We'll plug in the values of n from 1 to 10 and add up the results. Here's how:
1. Calculate the sum of the first 10 terms:
(1 / (9 + 1^5)) + (1 / (9 + 2^5)) + ... + (1 / (9 + 10^5))
2. Plug in the values of n and calculate each term:
(1 / 10) + (1 / 19) + (1 / 34) + (1 / 65) + (1 / 126) + (1 / 241) + (1 / 466) + (1 / 901) + (1 / 1750) + (1 / 3401)
3. Add up all the terms:
Approximate sum = 0.1 + 0.0526315789 + 0.0294117647 + 0.0153846154 + 0.0079365079 + 0.0041493776 + 0.0021447721 + 0.0011109889 + 0.0005714286 + 0.0002946192 = 0.14494 (rounded to five decimal places)
Now, to estimate the error, we can compare the sum of the first 10 terms (0.14494) with the value of the integral of the series from 10 to infinity. Since we don't have the exact formula for the integral, we can't calculate it directly. However, we can use the Remainder Estimation Theorem, which states that the error, Rn, is less than or equal to the absolute value of the integral from n+1 to infinity. In this case, we're using n=10:
R10 = ∫[10 to infinity] (1 / (9 + x^5)) dx
To find the error, we need to integrate the series from 10 to infinity. Since this is not possible to do exactly, we'll have to estimate it using numerical methods or software. Unfortunately, we don't have enough information to provide an estimate for the error value in this case