Consider the convergent alternating series Summation from k equals 1 to infinity StartFraction (negative 1 )Superscript k Over (
2 k plus 1 )cubed EndFraction . If the third partial sum Upper S 3 is used to estimate the value of the series, what is the upper bound given by the Alternating Series Remainder Theorem on the absolute error |R 3 | in this estimation?
1 answer:
Answer:
Step-by-step explanation:
Check attachment for solution.
Let assume n = k, it does not change anything
Given the series
Σ(-1)ⁿ / (2n+1)ⁿ from n=1 to n= infinity
We can express the series as the sum of partial sums and infinite remainder
S = Sn + Rn
The partial sum is already set to the 3rd terms since the upper bound is given to be 3
So, the error should be from the 4th term to infinity.
NOTE: the error is dominated by the first term of the error. The first term of the error is the fourth term.
Therefore,
The error bound = the value of the fourth term
So, when n = 4
Rn = (-1)ⁿ / (2n + 1)³
R4 = (-1)⁴ / (2•4 + 1)³
R4 = 1 / (8 + 1)³
R4 = 1 / 9³
Rn = 1 / 729
The error bound is positive
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