Approximate the sum of the series correct to four decimal places.
n=1 to infinity
(-1)^n/4^n*n!
(-1)^1/4^1*1 + (-1)^2/4^2*2 + (-1)^3/4^3*3 + (-1)^4/4^4*4 + (-1)^5/4^5*5 + (-1)^6/4^6*6
To approximate the sum of this series, we can use mathematical software like Wolfram Alpha or Python programming language.
Using Wolfram Alpha:
1. Open a web browser and go to the Wolfram Alpha website (www.wolframalpha.com).
2. In the search bar, type "sum of (-1)^n/4^n*n! from n=1 to infinity" and hit Enter.
3. Wait for Wolfram Alpha to compute the result.
4. The result will be shown as an approximate value. Round the result to four decimal places.
Using Python programming language:
1. Open a Python IDE or start a Python interpreter.
2. Define a function for the series expression:
```python
import math
def series_term(n):
return (-1)**n / (4**n * math.factorial(n))
```
3. Write a loop to calculate the partial sums of the series:
```python
sum = 0
n = 1
approximation = 0
while True:
term = series_term(n)
sum += term
n += 1
if abs(term) < 1e-5:
approximation = round(sum, 4)
break
```
4. Print the approximation:
```python
print(approximation)
```
Running this code will calculate the partial sums of the series until the absolute value of the term becomes smaller than 1e-5 (which means the approximation is accurate to four decimal places) and then round the sum to four decimal places.