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 need to estimate the difference between the actual value and the approximation given.
The given series is -0.1826666..., which can be written in mathematical notation as:
S = ∑((-1)^(n+1))/(n*5^n), where n ranges from 1 to infinity.
To estimate the error, we can use the nth-term test for series convergence. According to this test, if the series converges, then the nth term of the series approaches zero as n approaches infinity. In our case, the nth term can be written as:
|(-1)^(n+1)|/(n*5^n) = 1/(n*5^n)
Now, let's find the upper bound of the error by finding the sum of the remaining terms after a certain point (n = k) and taking the absolute value.
|Error| = |Actual value - Approximation|
|Error| = |S - S(k)|, where S(k) is the partial sum of the series up to the kth term.
To calculate S(k), we can use the formula for the sum of a convergent geometric series:
S(k) = a(1 - r^k)/(1 - r),
where a is the first term and r is the common ratio. In our case, a = 1/5 and r = -1/5.
S(k) = (1/5)(1 - (-1/5)^k)/(1 - (-1/5)).
Now, plug the value of S(k) into the error formula:
|Error| = |S - S(k)| = |S - [(1/5)(1 - (-1/5)^k)/(1 - (-1/5))]|
Please note that since we are dealing with an infinite series, it is not possible to find the exact value of the error. Instead, we can use the formula above to find an upper bound on its absolute value.