Let and be two sequences that satisfy the relationship
for all .
- Prove that if for all then for all .
- Prove that if for all and if is convergent, then
- Proof. Assume for all . From the given relation between the and we have
Now, we claim for all . To see this, consider the function
Since for all we have is increasing for all . Since we must have for all . Since by assumption we then have
But, this implies
- Proof. To prove converges we use the limit comparison test with . First, since converges we know
Now, we use the given relation between the and the ,
Then, use the expansion of the exponential,
This gives us
Therefore, by the limit comparison test the series and either both converge or both diverge. Since converges by hypothesis, we have established the convergence of