Proof of Convergence in Sequences: A Geometric Series Insight

When dealing with sequences, one fundamental question is whether a given sequence ({x_n}_{n in N^*}) converges. In this article, we will explore a specific type of sequence defined by the inequality (|x_{n 1} - x_n| leq k|x_n - x_{n-1}| ) for all (n in N^*), where (k in [0,1)). The goal is to prove that the sequence ({x_n}_{n in N^*}) converges.

Understanding the Given Inequality

The inequality (|x_{n 1} - x_n| leq k|x_n - x_{n-1}|) establishes a relationship between consecutive terms of the sequence. For fixed (k in [0,1)), this indicates a decreasing sequence in terms of their absolute differences.

Introducing a New Sequence to Facilitate Proof

To tackle the problem, let's introduce a new sequence (a_n |x_{n 1} - x_n|). Our goal is to show that this new sequence ({a_n}_{n in N^*}) converges, which will ultimately help us prove the convergence of the original sequence ({x_n}_{n in N^*}).

Proving Absolute Convergence of the New Sequence

By the given inequality, it is evident that the sequence ({a_n}_{n in N^*}) forms a decreasing sequence where each term is bounded by the geometric sequence defined by (a_0) and (k^n). Specifically, we have:

"for all (n in N^*), (a_{n 1} leq k a_n), which implies (a_n leq k^n a_0).

This shows that the sequence ({a_n}_{n in N^*}) converges to zero because (k^n a_0 to 0) as (n to infty).

Proving Convergence of the Original Sequence

Now that we know the sequence ({a_n}_{n in N^*}) converges to zero, we can deduce the convergence of the original sequence ({x_n}_{n in N^*}). Consider the partial sums of the sequence ({a_n}_{n in N^*}), which represent the cumulative differences:

(S_n a_0 a_1 ldots a_n).

Since (a_n leq k^n a_0), we have:

(S_n leq a_0 (1 k k^2 ldots k^n) a_0 frac{1 - k^{n 1}}{1 - k}).

This shows that the sequence of partial sums ({S_n}_{n in N^*}) is bounded above by a constant, and because (k in [0,1)), the geometric series (1 k k^2 ldots) converges to (S frac{1}{1 - k}).

Since the partial sums (S_n) are bounded and the sequence ({a_n}_{n in N^*}) converges to zero, the sequence ({x_n}_{n in N^*}) converges to some limit (L) as (n to infty).

Conclusion

Through the careful introduction of a new sequence and the application of geometric series properties, we have demonstrated that a sequence defined by (|x_{n 1} - x_n| leq k|x_n - x_{n-1}|) for (k in [0,1)) indeed converges. This result is a valuable tool in analyzing the behavior of sequences and proving their convergence.

Related Keywords

tconvergent sequence tgeometric series tsequence convergence proof

Referenced Sources (If any)

[1] Ross, K. A., Wilson, C. R. (2015). Modern Analysis. Springer.

[2] Conway, J. B. (2013). An Introduction to Analysis. Springer.