Combining power series Use the power series representation
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Question

Combining power series Use the power series representation

f(x ) =ln (1 − x) = −∑ₖ₌₁∞ xᵏ/k, for −1 ≤ x < 1,

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

p(x) = 2x⁶ ln(1 − x)

Accepted Answer

Given g of Z equals natural log of 1 minus Z, which equals the negative sum from 1 to infinity of z m divided by m or -1 equals Z 1, find the power series for R Z equals -4 z2 natural log of 1 minus c centered at 0 and give the interval of convergence. Let's go ahead and substitute our series. So we noticed our original series is just -4 Z2 multiplied by natural log of 1 minus Z. We can say we have -4 Z2d as multiplied by our sum negative sum from M equals 1 to infinity. Of z to the m divided by m. Now, we can actually combine our exponents here. Z squared multiplied by Z to the M. We can rewrite our sum As M equals 1 to infinity, which is now positive due to the negatives canceling. Of 4 Z raise to the M + 2. Divided by m Now from here, let's reindex our sun. Well, first, let's. In Equals M + 2. That means then, when M equals 1, N equals 3. Our new sum now is the sum from N equals 3 to infinity. A 4 Z raise to the N. Divided by N minus 2. This is because, if N equals M + 2, M equals N minus 2. Now, let's find our interval of convergence, since we have our power series. So We'll say, first as Z equals 1. As he equals 1, the sum. As N equals 3 to infinity. Goes to 4. Divided by n minus 2. We would get one race to the end, which is always one. Now we can rewrite this again. This is equivalent to 4 multiplied by the sum, as M equals 1 to infinity. Of 1 divided by M. This is the harmonic series, and we know that by the P series, this diverges. Meaning as Z equals 1, we will diverge. Now? Let's use Z equals -1. So, if we substitute the spec in, We get the series. As n equals 3 to infinity of 4 divided by n minus 2 multiplied by -1 raised to the n power. We will then use the alternating series test to determine this convergence. We will let B sub m. Which in our case is actually B in equals. 4 divided by. In minus 2. Now the limit of B subn should Equals 0. And be subn should decrease. If these are true, then we will converge. Let's go ahead and check. We have 4 divided by N minus 2. Now, this is a decreasing sequence, because it's 4 divided by N minus 2. And this does approach 0. Meaning that our series converges. As Z equals -1. We can then find their interval of convergence. Which we notice. -1 is convergent, which means it gets a square bracket, and 1 is divergent, which gives an open bracket. Giving us our interval of convergence. OK, I hope to help you solve the problem. Thank you for watching. Goodbye.

Combining power series Use the power series representation

f(x ) =ln (1 − x) = −∑ₖ₌₁∞ xᵏ/k, for −1 ≤ x < 1,

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

p(x) = 2x⁶ ln(1 − x)

Key Concepts

Power Series Representation

Interval of Convergence

Operations on Power Series

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