Alternating Series Remainder Theorem Worksheet
Practice problems and explanations for the Alternating Series Remainder Theorem in Calculus.
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Alternating Series Remainder Theorem
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The Alternating Series Remainder Theorem states that for a convergent alternating series Σ (-1)^n * a_n, where a_n > 0, the absolute value of the remainder R_n (the error in approximating the sum S by the nth partial sum S_n) is less than or equal to the absolute value of the first neglected term, a_(n+1). That is, |R_n| ≤ a_(n+1).
1. Which of the following conditions is NOT required for the Alternating Series Remainder Theorem to apply?
The series must be alternating.
The terms a_n must be positive.
The series must converge absolutely.
The terms a_n must be decreasing.
2. For an alternating series Σ (-1)^n * a_n, the error in approximating the sum by the nth partial sum is denoted by , and its absolute value is always less than or equal to the absolute value of the term.
3. Consider the alternating series Σ (from n=1 to ∞) (-1)^(n+1) / n. If you approximate the sum of this series using the first 4 terms, what is the maximum possible error according to the Alternating Series Remainder Theorem?
4. The Alternating Series Remainder Theorem can be used for any alternating series, regardless of convergence.
True
False
5. Given the series Σ (from n=0 to ∞) (-1)^n / (n!):
a) Write out the first 5 terms of the series.
b) Approximate the sum of the series using the first 4 terms.
c) Use the Alternating Series Remainder Theorem to find an upper bound for the error in your approximation from part (b).