Identify two different divergent sequences a and b who difference a - b converges.
To identify two different divergent sequences a and b whose difference a - b converges, we need to use some algebraic manipulation. Let's break it down step by step.
First, let's consider two divergent sequences a[n] and b[n] such that a[n] diverges to positive infinity and b[n] diverges to negative infinity. This ensures that the sequences are divergent but still allows their difference to converge.
For sequence a[n], we can use the sequence of positive integers: a[n] = n, where n is a positive integer (e.g., 1, 2, 3, ...).
For sequence b[n], we can use the sequence of negative integers: b[n] = -n, where n is a positive integer (e.g., 1, 2, 3, ...).
Now, let's find the difference between a[n] and b[n], denoted as a[n] - b[n]:
a[n] - b[n] = n - (-n) [Substituting the values of a[n] and b[n]]
= n + n [Negation of a negative number]
= 2n [Simplification]
As n approaches infinity, the value of 2n also approaches infinity. However, the difference 2n is a constant multiple of n, which means it converges to positive infinity.
So, we have successfully identified two different divergent sequences a[n] = n and b[n] = -n, whose difference a[n] - b[n] = 2n converges to positive infinity.