Errors In Using Convergence Tests In Infinite Series

Murat Genç,Mustafa Akinci

doi:10.24193/adn.13.2.8

Abstract

Abstract: The present study aimed to identify the errors made by pre-service elementary mathematics teachers while investigating the convergence of infinite series. A qualitative exploratory case study design was used with a total of 43 undergraduate students. Data were obtained from a test administered in a paper-and-pencil form consisting of seven open-ended questions. The data analysis was done using descriptive and content analysis techniques. Findings were presented as follows: inappropriate test selections; failure to check convergence criteria; incorrect use of a comparison test; limit comparison test error; re-test convergence test results; considering ∑ as a multiplicative function; misunderstanding of special series; considering that series has no character when the convergence test is inconclusive; confusing sequences with series; misunderstanding of the nth-term test; misinterpretation of convergence test results. Findings showed that students with insufficient procedural knowledge had difficulty in solving the given problem even if they understood it, whereas those with insufficient conceptual knowledge could not literally understand what they did even if they solved the problem. Therefore, the establishment of a moderate balance between procedural and conceptual knowledge in the learning of the convergence of series is essential in reducing the errors or learning difficulties for developing deep mathematical understanding

Highlights

In today’s developing world, mathematics in many areas is becoming increasingly important for individuals, communities, and nations to function across the globe
2 Failure to Check Convergence Criteria of Tests Another error was that participants used convergence tests regardless of the characteristics that the series should have in the criteria of the convergence tests
In this study, it was attempted to explore the types of errors that elementary mathematics pre-service teachers make when examining the characters of the series

Summary

Introduction

In today’s developing world, mathematics in many areas is becoming increasingly important for individuals, communities, and nations to function across the globe. The introduction to calculus begins with the concept of limit, but this concept is inherently difficult, and no matter how it is taught many students have several difficulties in understanding the definition of limit intuitively (Barnes, 1995). These difficulties mostly stem from the inability to understand the notion of infinity or a fairly complex definition of the limit conceptually (Cornu, 1991, Tall, 1992; Tall & Vinner, 1981). Many students experience similar difficulties in understanding the convergence of an infinite series which is one of the important concepts of analysis (Akgün & Duru, 2007). These difficulties have been said to increase further along with the inadequacy in algebra (Brown, 1996)

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