Finding the values of the variable where a series converge
The sum of the series is S = 20+80x^2 + 320x^3 +1280x^4+..... rewritten as S = 20 / (1 - 4x⁻¹) converges for he values |x⁻¹| < 1/4
To obtain what values of the variable the series converge and use the properties of the geometric series to find the sum of the series when it converges, we have to first look at the series given: 20 + 80x² + 320x³ + 1280x⁴ + .........Let's take the first three terms of the series: a₁ = 20a₂ = 80x²a₃ = 320x³
To find the ratio r, let's divide the second term by the first and the third term by the second:a₂/a₁ = (80x²)/20 = 4x²a₃/a₂ = (320x³)/(80x²) = 4x We notice that the ratio r is constant and equal to 4x⁻¹.
Now we can determine the conditions for which the series converges. Condition for the convergence of a geometric series:|r| < 1|r| = |4x⁻¹| < 1|r| = 4|x⁻¹| < 1|r| = |x⁻¹| < 1/4 Therefore, the domain of the series is the set of all x for which |x⁻¹| < 1/4. Since x cannot be zero, we exclude x = 0 from the domain. Then, the domain is obtained by taking the reciprocal of the inequality and inverting the sign:|x⁻¹| < 1/4⇔ - 1/4 < x < 1/4⇔ x ∈ (-1/4, 1/4)
Finally, we can use the formula for the sum of a finite geometric series to find the sum of the given series:S = a₁ (1 - rⁿ) / (1 - r)where n is the number of terms in the sum. Since we have an infinite series, we can take the limit as n approaches infinity:S = a₁ / (1 - r) = 20 / (1 - 4x⁻¹) for |x⁻¹| < 1/4Therefore, the sum of the series is S = 20 / (1 - 4x⁻¹) when the series converges for |x⁻¹| < 1/4.