bballninja said:I'm pretty sure it's true, since each successive term of the sequence will be larger or equal to the previous, so xn/xn+1 should always be ≤ 1
bballninja said:Well would I be able to claim that it is non-decreasing monotone, and show that it is bounded which implies convergence?
bballninja said:Ahh I'm so sorry haha. The instructor just emailed us that there was a typo and the ratio should actually be xn / (xn+1). This makes more sense now. Thanks for your help though
A convergence proof is a type of mathematical proof that demonstrates that a sequence of numbers or functions approaches a specific value or limit as its inputs increase. It is often used in calculus and other areas of mathematics to show the behavior of a sequence or series.
A convergence proof is important because it provides a rigorous and logical proof of the behavior of a sequence or series. This can help to confirm the validity of mathematical theories and provide a deeper understanding of the behavior of functions.
The key steps in a convergence proof typically involve showing that the sequence or series is bounded, monotonic, and that the limit of the sequence or series is the desired value. This is often done using algebraic manipulations and the definition of limits.
Yes, there are different types of convergence proofs depending on the type of sequence or series being analyzed. Some common types include convergence by comparison, convergence by ratio or root test, and convergence by the squeeze theorem.
To improve your skills in writing convergence proofs, it is important to have a strong understanding of mathematical concepts such as limits and series. Additionally, practice and reviewing examples of convergence proofs can help improve your skills over time.