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Remainders and estimates Consider the
Chapter 10, Problem 36E(choose chapter or problem)
35-42. Remainders and estimates Consider the following convergent series.
a. Find an upper bound for the remainder in terms of n.
b. Find how many terms are needed to ensure that the remainder is less than \(10^{-3}\).
c. Find lower and upper bounds (\(L_{n}\) and \(U_{n}\), respectively) on the exact value of the series.
d. Find an interval in which the value of the series must lie if you approximate it using ten terms of the series.
\(\sum_{k=1}^{\infty} \frac{1}{k^{8}}\)
Questions & Answers
QUESTION:
35-42. Remainders and estimates Consider the following convergent series.
a. Find an upper bound for the remainder in terms of n.
b. Find how many terms are needed to ensure that the remainder is less than \(10^{-3}\).
c. Find lower and upper bounds (\(L_{n}\) and \(U_{n}\), respectively) on the exact value of the series.
d. Find an interval in which the value of the series must lie if you approximate it using ten terms of the series.
\(\sum_{k=1}^{\infty} \frac{1}{k^{8}}\)
ANSWER:SOLUTION
The given convergent series is
Step 1
(a).
Find an upper bound for the remainder in terms of n.
Let be the remainder of the given series.
The upper bound of the remainder is given by
Where
Therefore