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Worked problem · Calculus II · Unit 4B

Interval Of Convergence Example: Worked Example and Solution

A complete taylor or power-series construction example with method choice, derivation, verification, and a common wrong approach.

1 problem8 min estimated timeIntermediate to advanced progression

Problem

  1. Find the interval of convergence of ∑n=1∞xn/n\sum_{n=1}^{\infty} x^n/n.

Complete worked solutions

Every problem has a source-matched answer and independently reviewed derivation.

01

Problem 1: Find the interval of convergence of ∑n=1∞xn/n\sum_{n=1}^{\infty} x^n/n.

Answer: [−1,1)[-1,1)

Why this method: Taylor or power-series construction matches the mathematical structure before any algebraic cleanup.

  1. The radius is 1. At x=1x=1 the harmonic series diverges; at x=−1x=-1 the alternating harmonic series converges.
  2. Therefore the interval is [−1,1)[-1,1).

What is included

See a concise answer and full derivation for find the interval of convergence of sum from n equals 1 to infinity x to the power (n) / n.

Skills assessed

  • Taylor or power-series construction

Prerequisites

  • derivatives
  • infinite series
Interval Of Convergence Example instructional sequence
A complete taylor or power-series construction example with method choice, derivation, verification, and a common wrong approach. The numbered labels and written sequence preserve meaning without relying on color.
Long description

Read the diagram from top to bottom. Each numbered box names one decision or mathematical operation. Arrows show the required order; the text labels remain the complete interpretation in print, dark mode, and nonvisual reading.

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Common errors

  • Transforming the formula but forgetting to transform the interval, endpoints, factorial, or remainder.

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Supporting steps