Combining power series Use the geometric series

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Question

Combining power series Use the geometric series

f(x) = 1/(1-x) = ∑ₖ₌₀∞ xᵏ, for |x| < 1,

to find the power series representation for the following functions (centered at 0). Give the interval of convergence of the new series.

f(x³) = 1/(1 − x³)

Accepted Answer

Find the power series representation for PF X equals 1 divided by 1 minus X to the fourth. center at 0 and give the interval of convergence. Now, to solve this, let's first notice our geometric series. 1 divided by 1 minus R is equivalent to the sum. As in equals 0 to infinity. Of our race to the end. For the absolute value of R is less than 1. We can actually modify ours. We have 1 divided by 1 minus X to the 4th, which is the same as saying the sum as it equals 0 to infinity. Of x to the 4th raised to the end power. We can simplify even further to get the sum from N equals 0 to infinity of X-rays to the 4 N. Now let's find our convergence. We know it converges for the absolute value of R less than 1, which means we have the absolute value of X to the 4th is less than 1. This just then means the absolute value of X is less than 1. Let's check our end points now. We will use the divergence test. This just tells us if the limit. As in approaches infinity. Of the series is not equal to 0. Or it doesn't exist. The series diverges. Let's check our end points. Since we have the absolute value of X less than 1, we'll first take X equals 1. We get the sum As N equals 0 to infinity. Of one raised to the 4 in. This will be equivalent So one Which means The limit, as in the approaches infinity, will be equals to one. This is not equals 0, which means this series diverges, 4 X equals 1. Let's check the same, but for negative one. X equals -1 by the absolute value. We have the sum as in equals 0 to infinity. Of -1 raised to the 4 in. No. The negative one Alternates, which means our limit. does not exist, which means this is also divergent. This means our series diverges at both ends, which means our interval of convergence. Will be from native 1 to 1. And since we have a less than, and not a less than or equals to. We will use parentheses. As well as because it diverges at those points. So we have our interval of convergence, and we have our series itself. OK, I hope to help you solve the problem. Thank you for watching. Goodbye.

Combining power series Use the geometric series

f(x) = 1/(1-x) = ∑ₖ₌₀∞ xᵏ, for |x| < 1,

to find the power series representation for the following functions (centered at 0). Give the interval of convergence of the new series.

f(x³) = 1/(1 − x³)

Key Concepts

Geometric Series and Its Power Series Representation

Substitution in Power Series

Interval of Convergence

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