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Higher Mathematics

Infinite Series

This chapter systematically studies the basic concepts of infinite series, convergence tests, power series, Taylor series, and their applications.

Basic Concepts of Infinite Series

  • Definition of a series: ∑n=1∞an\sum_{n=1}^{\infty} a_n
  • Partial sums, convergence and divergence
  • Sum of a convergent series

Convergence Tests

  • Series with positive terms: comparison test, ratio test, root test
  • Alternating series: Leibniz test
  • Arbitrary series: absolute and conditional convergence

Common Series

  • Geometric series: ∑arn\sum ar^n
  • pp-series: ∑1np\sum \frac{1}{n^p}

Power Series

  • Form: ∑n=0∞an(x−x0)n\sum_{n=0}^{\infty} a_n (x - x_0)^n
  • Radius and interval of convergence
  • Properties: termwise differentiation and integration

Taylor Series

  • Taylor formula, Taylor expansion
  • Convergence and applications

Exercises

  1. Determine the convergence of the series ∑n=1∞1n2\sum_{n=1}^{\infty} \frac{1}{n^2} and find its sum.
  2. Determine the convergence of the series ∑n=1∞(−1)n+1n\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n}.
  3. Find the radius and interval of convergence of the power series ∑n=0∞xn\sum_{n=0}^{\infty} x^n.
  4. Write the Taylor expansion of exe^x.
  5. Determine the convergence of the series ∑n=1∞n!nn\sum_{n=1}^{\infty} \frac{n!}{n^n}.
Reference Answers

1. Determine the convergence of ∑n=1∞1n2\sum_{n=1}^{\infty} \frac{1}{n^2} and find its sum

pp-series, p=2>1p=2>1, converges. The sum is π26\frac{\pi^2}{6}.

2. Determine the convergence of ∑n=1∞(−1)n+1n\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n}

Alternating series, satisfies the Leibniz test, converges.

3. Find the radius and interval of convergence of the power series ∑n=0∞xn\sum_{n=0}^{\infty} x^n

Geometric series, converges for ∣x∣<1|x|<1, radius R=1R=1, interval (−1,1)(-1,1).

4. Write the Taylor expansion of exe^x

ex=∑n=0∞xnn!e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!}

5. Determine the convergence of ∑n=1∞n!nn\sum_{n=1}^{\infty} \frac{n!}{n^n}

Ratio test: lim⁡n→∞an+1an=0\lim_{n\to\infty} \frac{a_{n+1}}{a_n} = 0, so it converges.