convergent sequence

USAGE SUMMARY

The phrase "convergent sequence" is correct and usable in written English. You can use it in mathematical contexts, particularly in discussions about limits and series. For example: "The terms of the convergent sequence approach a specific limit as n increases." Alternative expressions include "converging sequence" and "limit sequence."

✓ Grammatically correct

Science

Human-verified examples from authoritative sources

Exact Expressions

60 human-written examples

(ii) Any subsequence of a convergent sequence converges to the same limit.

Fixed Point Theory and Applications

This first attempt to detect genome-wide convergent sequence evolution across divergent taxa reveals the phenomenon to be much more pervasive than previously recognized.

Nature

Definition 4.1 (Convergent sequence).

Journal of Inequalities and Applications

Any convergent sequence is bounded.

Fixed Point Theory and Applications

Therefore, is a convergent sequence.

Journal of Inequalities and Applications

Let be a convergent sequence in.

Fixed Point Theory and Applications

Any convergent sequence has a unique limit.

Fixed Point Theory and Applications

convergent sequence of always belongs to.

Fixed Point Theory and Applications

(iii) Any convergent sequence is bounded.

Fixed Point Theory and Applications

(1) Let be a convergent sequence in.

Fixed Point Theory and Applications

Every statistically convergent sequence is statistically bounded.

Journal of Inequalities and Applications

Expert writing Tips

Best practice

When using the term "convergent sequence", ensure you've established the context of a mathematical or analytical discussion. It implies a specific type of sequence with a defined limit.

Common error

Avoid assuming that a bounded sequence is automatically a "convergent sequence". Boundedness is a necessary, but not sufficient, condition for convergence. A sequence must approach a specific limit to be considered convergent.

Linguistic Context

The phrase "convergent sequence" functions as a noun phrase, specifically referring to a sequence that approaches a finite limit. As Ludwig AI confirms, this phrase is commonly used in mathematical contexts.

Expression frequency: Very common

Frequent in

Science

100%

Less common in

News & Media

0%

Formal & Business

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Ludwig's WRAP-UP

In summary, "convergent sequence" is a noun phrase used primarily in mathematics and science to describe a sequence that approaches a finite limit. Ludwig AI confirms its correct grammatical status and high frequency in specialized literature. While fundamentally correct, remember that boundedness does not guarantee convergence. For alternatives, you might consider phrases like "converging series" or "sequence with a limit", depending on the specific nuance you want to convey.

More alternative expressions

Phrases that express similar concepts, ordered by semantic similarity:

converging series

Focuses on the series aspect rather than the general sequence, implying an ordered list of terms summed together that approach a limit.

sequence with a limit

Emphasizes the existence of a limit for the sequence, diverging from just stating convergence.

limited sequence

Highlights the boundedness and the existence of a limit, rather than the process of approaching it.

sequence approaching a limit

Focuses on the dynamic process of the sequence getting closer to its limit.

sequence tending to a limit

Similar to 'approaching', but suggests a mathematical 'tendency'.

accumulation sequence

Highlights the accumulation aspect near a certain point, but not necessarily convergence in the strict sense.

Cauchy sequence

Describes a sequence where the terms become arbitrarily close to each other, a condition related to, but not identical to, convergence.

asymptotic sequence

Focuses on the behavior as the index approaches infinity, highlighting the asymptotic nature rather than the convergence itself.

null sequence

Specifically refers to a sequence that converges to zero.

bounded sequence with a limit point

Highlights that, even though the sequence has a limit point, it might not necessarily converge to it.

FAQs

How is "convergent sequence" used in mathematics?

In mathematics, a "convergent sequence" is a sequence whose terms approach a limit. This concept is fundamental in calculus and analysis.

What are some properties of a "convergent sequence"?

A "convergent sequence" is bounded and has a unique limit. Any subsequence of a convergent sequence also converges to the same limit.

Is every bounded sequence also a "convergent sequence"?

No, not every bounded sequence is a "convergent sequence". A bounded sequence may oscillate and not approach a specific limit, while a convergent sequence must approach a finite limit.

What is the difference between a "convergent sequence" and a "divergent sequence"?

A "convergent sequence" approaches a limit, while a "divergent sequence" does not have a finite limit or oscillates indefinitely.