Simplify:
2/1! - 3/2! + 4/3! - 5/4! + ... + 2010/2009! - 2011/2010!, where k! denotes the product of all integers from 1 up to k, that is,
1! = 1,
2! = 1x2 = 2,
3! = 1x2x3 = 6, and so on.
To simplify the given expression 2/1! - 3/2! + 4/3! - 5/4! + ... + 2010/2009! - 2011/2010!, we need to understand the pattern and evaluate each term.
Let's break it down step by step:
Step 1: Determine the pattern.
The given expression has a pattern where the numerator increases by 1 and the denominator increases by 1 factorial (n!). The signs alternate between positive and negative.
Step 2: Write out the terms.
We write out the first few terms to see the pattern:
2/1! - 3/2! + 4/3! - 5/4! + ...
Step 3: Evaluate each term.
Let's evaluate the first few terms:
2/1! = 2/1 = 2
3/2! = 3/(1x2) = 3/2 = 1.5
4/3! = 4/(1x2x3) = 4/6 = 2/3
5/4! = 5/(1x2x3x4) = 5/24
We can observe that after simplifying, the terms are getting smaller. Let's continue evaluating until we see a pattern:
5/4! = 5/(1x2x3x4) = 5/24 ≈ 0.208
6/5! = 6/(1x2x3x4x5) = 6/120 ≈ 0.05
7/6! ≈ 0.0104
...
Step 4: Identify the pattern and simplify the expression.
From our evaluations, we see that the terms become smaller as we progress. Eventually, the terms become negligible, and we can approximate the sum as a constant value.
Observing the denominators of each term (1!, 2!, 3!, 4!, ...), we can generalize the pattern of the nth term as n/(n-1)!
So we can rewrite the given expression as:
Sum of (n/(n-1)!) for n = 2 to 2011
Now we can approximate the sum as follows:
2/1! - 3/2! + 4/3! - 5/4! + ... ≈ 2 - 1.5 + 2/3 - 0.208 + 0.05 - 0.0104 + ...
As we progress to larger terms, the values become negligible, and we can approximate the sum as:
2 - 1.5 + 2/3 - 0.208 + 0.05 - 0.0104 ≈ 1.6396
Therefore, the simplified value of the given expression is approximately 1.6396.