Power Series Math Example 1

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Example 1

medium
Find the radius of convergence of ∑n=0∞xnn+1\displaystyle\sum_{n=0}^{\infty} \frac{x^n}{n+1}.

Solution

  1. 1
    Apply the ratio test: compute ∣an+1an∣\left|\frac{a_{n+1}}{a_n}\right| where an=xnn+1a_n = \frac{x^n}{n+1}.
  2. 2
    Simplify the ratio: ∣xn+1n+2⋅n+1xn∣=∣x∣⋅n+1n+2→n→∞∣x∣\left|\frac{x^{n+1}}{n+2} \cdot \frac{n+1}{x^n}\right| = |x|\cdot\frac{n+1}{n+2} \xrightarrow{n\to\infty} |x|
  3. 3
    The series converges when L=∣x∣<1L = |x| < 1, so the radius of convergence is R=1R = 1.

Answer

R=1R = 1
The ratio test extracts RR by finding the limiting ratio. Here L=∣x∣L = |x|, so R=1R=1.

About Power Series

An infinite series of the form ∑n=0∞an(x−c)n=a0+a1(x−c)+a2(x−c)2+⋯\sum_{n=0}^{\infty} a_n(x-c)^n = a_0 + a_1(x-c) + a_2(x-c)^2 + \cdots where cc is the center and ana_n are the coefficients. A power series defines a function of xx wherever it converges.

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More Power Series Examples

Example 2 hard

Find the interval of convergence of [formula].

Example 3 easy

Find the radius of convergence of [formula].

Example 4 medium

Differentiate [formula] term by term to find a new series identity.

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