Sum of the Series Sigma n=1 to infinity 1/(n-1)!
There is a series given as Sigma n=1 to infinity 1/(n-1)!. Let's break down the explanation into key points:
- Understanding the Series:
- The given series is 1/(n-1)! which represents the reciprocal of factorial of (n-1).
- The factorial of a number n is denoted by n! and is the product of all positive integers up to n.
- Calculating the Terms of the Series:
- When n=1, the term becomes 1/(1-1)! = 1/0! = 1/1 = 1.
- When n=2, the term becomes 1/(2-1)! = 1/1! = 1/1 = 1.
- When n=3, the term becomes 1/(3-1)! = 1/2! = 1/2 = 0.5.
- This pattern continues for all n values.
- Finding the Sum of the Series:
- As we sum up all the terms of the series from n=1 to infinity, we can observe that the series is actually a geometric series.
- The sum of an infinite geometric series can be calculated using the formula S = a/(1-r), where 'a' is the first term and 'r' is the common ratio.
- In this case, the first term is 1 and the common ratio is 1. Therefore, the sum of the series is S = 1/(1-1) = 1/0 = undefined.
Hence, the sum of the series Sigma n=1 to infinity 1/(n-1)! is undefined as it diverges.