Representing functions by power series Identify the functions ...

Question

Representing functions by power series Identify the functions represented by the following power series.

∑ₖ₌₀∞ (xᵏ)/(2ᵏ)

Accepted Answer

Hi everyone, let's take a look at this practice problem. This problem says to find the function represented by the power series, which is the sum of M equal to 0 to an infinity of Z rate to the power M divided by 4 rates to the power M. So in order to find the function represented by this power series, we're going to actually rewrite our power series. So this is going to be equal to the sum of M equal to 0 to infinity of the quantity of Z divided by 4 in quantity raised to the power M. By writing it in this form, we see that we have a geometric series. And so we call for geometric series that if we have the sum of M equal to 0 to infinity. Of R rates to the power M, that this is just going to be equal to 1 divided by the quantity of 1 minus R in quantity if the absolute value of R is less than 1. Now, in our case, R is just going to be equal to Z divided by 4. So that means that our series here is going to be equal to 1 divided by the quantity of 1 minus Z divided by 4 in quantity. Now, to remove the fraction out of the denominator, we'll multiply both our numerator and denominator by 4. In doing so, we see that this is going to be equal to 4 divided by the quantity of 4 minus Z in quantity, and this here is the function that we are looking for. However, this function is only valid over a certain range of Z. And that's because for geometric series, the absolute value of R has to be less than 1. So, in our case, that means that the absolute value of Z divided by 4 has to be less than 1. So, multiplying both sides by 4, we get that the absolute value of Z has to be less than 4, or minus 4 is less than Z, which is less than 4. So we need to do is to check the end points of our interval here to see if our series converges at the end points. So we can begin by looking at Z equal to -4. And in this case, we will have the sum. Of M equal to 0 to infinity of the quantity of minus 4 divided by 4, in quantity raised to the power M, which we can simplify as being equal to the sum. Of M equal to 0 to infinity of the quantity of -1 and quantity raised to the power M. Now, since the limit as M approaches infinity of the quantity of 1 in quantity raised to the power M does not exist, that means that this series is going to diverge. So this series diverges, which means we will not include this end point in our interval of convergence. Next, we need to look at the case when Z is equal to 4. And in this case we'll have the sum of M equal to 0 to infinity of the quantity of 4 divided by 4 in quantity rates to the power M, which is just equal to the sum of M equal to 0 to infinity of 1 raised to the power M, which is equal to the sum of M equal to 0. To infinity of 1, since one race of the power M is always going to be equal to 1. Now this series also diverges. So, since this series diverges, that means we will not be including this end point in our interval of convergence. So that means that Z must be contained in the open interval between -4 and 4. So that is the second part of this problem that we are looking for. So just to recap, we found that our series is equal to 4 divided by the quantity of 4 minus Z in quantity. When Z is contained in the open interval from -4 to 4. So that is the answer that we're looking for for this problem. So I hope that this explanation has been useful, and I'll see you in the next video.

Representing functions by power series Identify the functions represented by the following power series.
∑ₖ₌₀∞ (xᵏ)/(2ᵏ)

Key Concepts

Power Series Representation

Geometric Series and Its Sum

Radius and Interval of Convergence

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