The geometric series 1 + [tex]\frac{1}{2} + \frac{1}{4} +\frac{1}{8}[/tex] +...... converging geometric series.
What is geometric series?
A geometric series is "the sum of the infinite number of terms that have common ratio between successive terms".
According to the question,
Geometric series ∑a rⁿ⁻¹ where 'a' is the coefficient and 'r' is the common ratio.
The converging geometric series depends on the value of 'r' if |r| < 1, then the geometric series converges. If |r|>1, then the geometric series diverges. If the |r| = 1, the geometric series diverges. If the |r| = -1, then the geometric series oscillates between 1 and -1.
- The series ∑ [tex]\frac{1}{5}2^(n-1)[/tex] has the common ratio 2 > 1, then the geometric series diverges.
- The series ∑7(-4)0ⁿ⁻1 has the common ratio -4 < 1, then the geometric series diverges.
- The series [tex]\frac{1}{81} + \frac{1}{27} + \frac{1}{9} +\frac{1}{3}+ ......[/tex] has the common ratio 1/27÷1/81 = 3, the common ratio 3 > 1, then the geometric series diverges.
- The series [tex]\frac{1} + \frac{1}{2} + \frac{1}{4} +\frac{1}{8}+ ......[/tex] has the common ratio 1/2÷1 = 1/2, the common ratio 1/2 < 1, then the geometric series diverges.
Hence, the series [tex]\frac{1} + \frac{1}{2} + \frac{1}{4} +\frac{1}{8}+ ......[/tex] has the common ratio 1/2÷1 = 1/2, the common ratio 1/2 < 1, then the geometric series diverges.
Learn more about geometric series here
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