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.

f(3x) = ln (1 − 3x)

Accepted Answer

Given the power C's representation, H of Y equals natural log of 1 minus Y, which equals the negative sum from 1 to infinity of Y N divided by N. And -1 is less equals to Y, which is less than 1. Find the power series for H of -4Y equals natural log of 1 + 4Y and state its interval of convergence. Now for our first task, we will find the series and we'll substitute. -4 Y for what? We'll substitute this in our series. We have negative, series. N equals 1 to infinity. Of -4 Y raised to the N divided by N. Now we can go ahead and manipulate this a little bit. We will get negative some. From one to infinity Of -4 race to the end. One reid in, why residian. And we actually put our negative with the 1. And this is all divided by in. Now, we do notice we have a negative outside of the sum, which can be combined into our -1 exponent. This gives us the positive sum now, as it equals one to infinity. A 4 to the inn -1 to the end plus 1. Why to the N divided by N. We can finally simplify this once more to get the sum as N equals 1 to infinity. Of -1 to the n + 1. 4 Y raise to the end all divided by n. This is our power series. Now let's find the interval of convergence for this power series. It told us why it's between -1 and 1. So let's copy that. We have -1, lesser equals 2. -4 Y, which is less than 1. Dividing by -4 on both sides gives us a new inequality. We have 1/4 is greater than or equals toy, which is greater than 1/4. Which we can rewrite To be negative 1/4 less than, less than or equals to 1/4. So now we have our values to check for our interval of convergence. Let's first let Y equals 1/4. Plugging this in gives us a sum as it equals 1 to infinity. Of -1, raised to the N + 1 multiplied by 1 ton. Defined by No. We can take the alternating series cysts to determine this convergence. For the alternating series tests, we need a series. That is a sub in. In a sub in a sub-series that is -1, raises the NBN. Or A n equals -1 to the n + 1. Be in Where bn is greater equals to zero for all in. If this is true, And if the limit as in approaches infinity of bn is equals a zero, and Be sub in is decreasing. We can tell That a subn is convergent. Now we know that our A sub N is going to be something of the form. 1 divided by N. This is always greater than 0. And it is decreasing and headed towards 0, meaning the limit. A sub N equals 0. This means by the alternating theory test. Our series Ape sub N is convergent. A Y equals 1/4. Now we can check Y equals -14. If we plug this in, we end up getting the series. As in equals one to infinity. Of -1 to the 2 n + 1. Divided by it. This just simplifies to give us this series, as it equals one to infinity. Of -1 divided by N. Or negative series and equals 1 to infinity. Of one divided by in Now, this series is a P series. This is actually the harmonic series, which means this diverges. So then we notice that Y equals 1/4 converges, Y equals -14 diverges, meaning our interval of convergence will be from -14 with a parenthesis to 1/4 with a square bracket. This is 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.

f(3x) = ln (1 − 3x)

Key Concepts

Power Series Representation

Substitution in Power Series

Interval of Convergence

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