convergent series

USAGE SUMMARY

'convergent series' is a grammatically correct and usable phrase in written English.
It refers to a sequence of numbers in which the terms become closer and closer to a finite limit, or a sequence of events that each contribute to achieving the same end result. For example, "The biologist studied the convergent series of traits found in the primate family."

✓ Grammatically correct

Science

Encyclopedias

News & Media

Human-verified examples from authoritative sources

Exact Expressions

60 human-written examples

In this technique, the solution is calculated in the form of a convergent series and convergence of the HPTM solutions to the exact solutions is shown.

Ain Shams Engineering Journal

I shall mean by a convergent series, one that converges on a finite, positive value, and by a non-convergent series, one that increases ad infinitum.

SEP

The method presents the solutions in terms of convergent series with easily computable components and improves the convergence of the series solution.

Advances in Difference Equations

The commutative law does not necessarily hold for multiplication of conditionally convergent series.

Encyclopedia Britannica

The method provides appropriate representation of the solutions in convergent series formula with accurately computable components.

Computers & Mathematics with Applications

We provide rapidly convergent series to compute these Clausen functions to high accuracy.

Journal of Computational Physics

Further, the limit n → ∞ yields conditionally convergent series defined by the Eisenstein summation method.

International Journal of Solids and Structures

An analytical closed-form solution has been obtained in terms of convergent series with easily computable components.

Procedia Engineering

Using the recent theory of Ern and Giovangigli, all the transport coefficients are expressed as convergent series.

Journal of Computational Physics

The solutions of our equation are calculated in the form of rabidly convergent series with easily computable components.

Journal of Computational Physics

The analytical solution is presented in the form of a convergent series, which is convenient to use.

Chemical Engineering Science

Expert writing Tips

Best practice

When discussing numerical methods, specify the type of convergence (e.g., absolute, uniform) to provide a more precise description of the series' behavior.

Common error

Ensure that you're accurately assessing whether a series converges before applying theorems or techniques that rely on convergence. Mistaking a divergent series for a convergent one can lead to incorrect results.

Linguistic Context

The phrase "convergent series" functions as a noun phrase, typically serving as the subject or object of a sentence when discussing mathematical or computational concepts. Ludwig's examples illustrate its use in scientific and engineering contexts.

Expression frequency: Very common

Frequent in

Science

85%

Encyclopedias

8%

News & Media

7%

Less common in

Formal & Business

0%

Wiki

0%

Reference

0%

Ludwig's WRAP-UP

The phrase "convergent series" is a grammatically sound and prevalent term, particularly within scientific and mathematical domains. As confirmed by Ludwig, it refers to an infinite series whose partial sums converge to a finite limit. Highlighting its widespread usage, "convergent series" appears frequently in academic papers, encyclopedias, and even news articles discussing scientific concepts. While alternatives like "converging sequence" or "series with a finite sum" exist, "convergent series" remains the standard and most precise terminology in mathematical discourse. When using this phrase, remember to be mindful of the specific type of convergence being discussed and to avoid confusing convergent series with divergent ones.

More alternative expressions

Phrases that express similar concepts, ordered by semantic similarity:

converging sequence

This term focuses on the ordered list of numbers rather than the sum of those numbers, but both lead to a finite limit.

series with a finite sum

This alternative describes the result of the series, emphasizing the bounded outcome.

absolutely convergent series

Adding "absolutely" specifies a stronger form of convergence, ensuring the series converges even when absolute values are considered.

uniformly convergent series

Adding "uniformly" specifies that convergence happens at the same rate for all points in the domain.

rapidly converging series

This alternative emphasizes the speed at which the series approaches its limit.

summable series

This term highlights the ability to compute a finite sum for the series.

finite series

This term means that the series has a limited number of terms.

limited sequence

This alternative focuses on the limited number of terms in the sequence.

series approaching a limit

It's a descriptive alternative that highlights the behavior of the series as it gets closer to a specific value.

bounded series

Bounded series has a upper or lower limit to the values it can take.

FAQs

How is a "convergent series" used in mathematical analysis?

In mathematical analysis, a "convergent series" is used to approximate functions, solve differential equations, and model physical phenomena. Its convergence allows for reliable calculations and meaningful interpretations.

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

A "convergent series" has a finite sum, meaning its partial sums approach a limit as the number of terms increases. A divergent series, on the other hand, does not have a finite sum; its partial sums do not approach a limit.

What are some real-world examples where "convergent series" are applied?

"Convergent series" find applications in signal processing, physics (e.g., calculating the motion of a damped oscillator), and computer science (e.g., approximating solutions to complex equations).

Can I use "converging sequence" instead of "convergent series"?

While related, "converging sequence" refers to an ordered list of numbers approaching a limit, whereas "convergent series" refers to the sum of those numbers also approaching a limit. The terms are often related but not perfectly interchangeable.