what is an upper bound on the absolute value of the error?
the abs value of (the sum from n=1 to infinity of ((-1)^(n+1))/(n*5^n) where it is -.1826666....)
To find an upper bound on the absolute value of the error, we can use the alternating series error bound formula. The formula states that the absolute value of the error, E, of an alternating series is less than or equal to the absolute value of the first neglected term.
In this case, the alternating series is defined as:
S = (-1)^(n+1) / (n*5^n)
The given value of S is approximately -0.1826666....
To find the upper bound on the absolute value of the error, we need to determine the absolute value of the first neglected term.
Since it is clear that the terms of the series are decreasing in magnitude, we can start by evaluating the term at the next value of n. In other words, let's calculate the value of n = 2:
S_2 = (-1)^(2+1) / (2*5^2) = -1/50 = -0.02
The first neglected term is approximately -0.02.
Therefore, the upper bound on the absolute value of the error is |E| ≤ |0.02| = 0.02.