Approximate the sum of the series correct to four decimal places.
(-1)^n/(n!5^n) (from n=1 to infinity)
You know that e^x = ∑x^n/n!
Now let x = -1/5
To approximate the sum of the given series to four decimal places, we'll have to calculate the terms of the series until the terms become very close to zero or until we reach the desired accuracy.
Let's start by calculating some terms of the series:
n = 1: (-1)^1 / (1! * 5^1) = -1/5
n = 2: (-1)^2 / (2! * 5^2) = 1/50
n = 3: (-1)^3 / (3! * 5^3) = -1/750
n = 4: (-1)^4 / (4! * 5^4) = 1/12000
n = 5: (-1)^5 / (5! * 5^5) = -1/187500
By observing the pattern, we can see that the magnitude of the terms is decreasing, but it's not decreasing quickly enough to simply sum a finite number of terms to achieve the desired accuracy.
In this case, we need to use an iterative method like the partial sum approximation method. We will start by calculating the sum of the first few terms and keep adding terms until the additional terms have a negligible impact on the final sum.
Let's apply the partial sum approximation method with calculation precision up to four decimal places:
Sum = (-1/5) + (1/50) - (1/750) + (1/12000) - (1/187500)
To determine when to stop adding terms, we'll need to set an error tolerance. Let's say our error tolerance is 0.0001.
We will keep adding terms until the absolute value of the next term becomes smaller than the error tolerance.
Let's perform the calculations until the error tolerance is satisfied:
Sum = -0.2 + 0.02 - 0.001333 + 0.00008333 - 0.0000053333
Sum ≈ -0.1992206667
Now, let's determine if the error is within our desired accuracy of four decimal places:
To do that, we need to calculate the absolute value of the next term:
n = 6: (-1)^6 / (6! * 5^6) = 1/31250000
The absolute value of the next term is 1/31250000 ≈ 3.2 × 10^(-8)
Since this value is smaller than our error tolerance of 0.0001 (10^(-4)), we can conclude that our approximation is accurate to four decimal places.
Therefore, the approximate sum of the given series, correct to four decimal places, is -0.1992.