Infinite Geometric Series Math Example 5

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Example 5

hard
For what values of xx does ∑n=0∞(x−1)n\displaystyle\sum_{n=0}^{\infty} (x-1)^n converge, and what is the sum?

Solution

  1. 1
    This is geometric with r=x−1r = x-1. Converges when ∣x−1∣<1|x-1| < 1, i.e., 0<x<20 < x < 2.
  2. 2
    Sum: 11−(x−1)=12−x\frac{1}{1-(x-1)} = \frac{1}{2-x}.

Answer

Converges for 0<x<20 < x < 2; sum =12−x= \dfrac{1}{2-x}
Treat x−1x-1 as the ratio rr. The convergence condition ∣r∣<1|r|<1 translates to ∣x−1∣<1|x-1|<1, giving the interval (0,2)(0,2).

About Infinite Geometric Series

The sum of all terms of a geometric sequence with common ratio ∣r∣<1|r| < 1. The infinite sum converges to a1−r\frac{a}{1-r}, where aa is the first term.

Learn more about Infinite Geometric Series →

More Infinite Geometric Series Examples

Example 1 easy

Find the sum of the infinite geometric series [formula].

Example 2 medium

Convert the repeating decimal [formula] to a fraction using an infinite geometric series.

Example 3 medium

Find the sum of the infinite series [formula]

Example 4 easy

Find the sum: [formula]

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