Series Convergence & Estimation Assessment Test

The correct answer is "Approached" because when a series converges, it means that the terms of the series get closer and closer to a certain number as the series progresses. This number is called the limit of the series. So, the terms of the series "approach" the limit value as the series goes on. Therefore, "Approached" is the appropriate word to describe the behavior of a number in a converging series.

A series can be defined as a sequence of sums, where each term is obtained by adding the previous terms in the sequence. It involves continuously adding numbers in a specific order, resulting in a sum at each step. This definition highlights the primary characteristic of a series, which distinguishes it from other options such as addition, collection of numbers, or line of figures. A series focuses on the cumulative sum of terms, rather than individual numbers or a visual representation.

A sequence which does not converge is said to be divergent. This means that the terms in the sequence do not approach a specific value or limit as the sequence progresses. Instead, the terms may fluctuate or diverge in different directions, indicating that the sequence does not have a well-defined limit.

The correct answer is "Limit of a sequence". In analysis, the concept of limit of a sequence is fundamental. It refers to the value that a sequence approaches as the terms of the sequence get closer and closer to a certain value. The limit of a sequence helps in understanding the behavior and properties of the sequence, and it is crucial in various areas of mathematics, such as calculus and real analysis.

A series that converges means that the sum of its terms approaches a specific value as the number of terms increases. This specific value is called the limit of the series. In this case, the correct answer is "Finite limit" because a convergent series has a limit that is a finite number, meaning it is not infinite or undefined.

A series that diverges means that the sum of its terms keeps getting larger and larger as more terms are added. In other words, the partial sums of the series approach infinity, indicating that there is no finite sum for the series.

A limit converges if it has a finite value. This means that as the input approaches a certain value, the output of the function approaches a specific number, rather than becoming infinitely large or undefined. In other words, the function "settles down" and approaches a fixed value as the input gets closer and closer to a certain point.

The interval of convergence refers to the range of values for which a series converges. It represents the set of all input values that will result in a convergent series. This interval can be determined using various methods such as the power rule or limiting power. Therefore, the correct answer is "Interval of convergence."

The correct answer is "Limit of a function." In calculus and analysis, the concept of the limit of a function refers to the behavior of the function as it approaches a specific input value. It helps determine if the function approaches a certain value or becomes infinite as the input approaches a given value. This concept is crucial in understanding the continuity, differentiability, and overall behavior of functions in calculus and analysis.

The correct answer is "Radius of convergence". This term refers to the interval in which a power series converges. It represents the distance from the center of the power series to the nearest point where the series converges. The radius of convergence determines the range of values for which the power series provides an accurate approximation of the function it represents.