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Infinite series

from class:

Algebra and Trigonometry

Definition

An infinite series is the sum of the terms of an infinite sequence. It can converge to a specific value or diverge to infinity.

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5 Must Know Facts For Your Next Test

  1. An infinite series converges if the sequence of partial sums approaches a finite limit.
  2. The geometric series $\sum_{n=0}^{\infty} ar^n$ converges if $|r| < 1$ and its sum is $\frac{a}{1 - r}$.
  3. The harmonic series $\sum_{n=1}^{\infty} \frac{1}{n}$ diverges.
  4. The ratio test can determine the convergence or divergence of an infinite series by evaluating $\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|$.
  5. The alternating series test states that an alternating series converges if its terms decrease in absolute value and approach zero.

Review Questions

  • What condition must be met for a geometric series to converge?
  • Explain why the harmonic series diverges.
  • How does the ratio test help in determining the convergence of an infinite series?
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