Suppose a function f is defined by the geometric series f(x) = (-1)kxk. k=0 a. Evaluate f(0) , f(0.3), f(0.8), f(1), and f(1.5), if possible b. What is the domain o...

Function defined by a series Suppose a function $f$ is defined by the geometric series $f(x)=\sum_{k=0}^{\infty} x^{2 k}$. a. Evaluate $f(0), f(0.2), f(0.5), f(1),$ and $f(1.5),$ if possible. b. What is the domain of $f ?$

Heard This problem we are given with geometric cities which is f or fix as it cools to sigma equals to zero in finite extra the Barkey on their do. Part of the question In the a part of the creation we have evaluate the function f of zero. Then if all point to then if off four point Fife on F off one. And finally if off 1.5 now from Terram 9.7 we known daughter Sigma cake worlds 20 toe in for night. Eight times are the bark A records to a divide by one minus are Where is the initial dome and our is though common ratio of the series. If you compare this given cities with the standard one, we can say that there are is X and value of is one so f zero and we know one conditional zits there that this formula is valid only win more X is sorry. Let me write correctly Mode are is less than one and this condition of getting used this formula otherwise, if more r is greater than one, then the series and diverges So now let noticed calculated the f of zero, So every zero will be quelled. Stew one divide by one minus are here is X and in the first case, X Harry zero. So one divided by one minus zero it comes out to Yes one. Next we're going to calculate F off 0.2. It will be equals to one divide by one minus X and x 0.2. So we'll get one divide by one minus 0.2, which comes out to be as 1.25 Actually going to calculate f off 0.5. It will be equal to one divide way one minus x on here x 0.5 So one divide by one minus 0.5 and that comes out will be asked to. Now we know that this family can only be valid when are is less than one. In our case, we can use this form. 11 more X is less than one now F off one Andi f off 1.5 will not be defined because more x less than one is not satisfied by X equals to one and X equals to 1.5. So finally we can compute. There F off one and F off 1.5 do not exist. This was the answer for a part of the question next to be sold in the bebop and in the be part, we have to tell it what is the domain of the function. So the remain off the function if is in the indoor pool, minus 1 to 1. So this was answer for the Beeb art.

Her in this problem were given with the geometric series, which is therefore fix is equal to two Sigma K equals 20 to infinite minus one toe the power key X time X to the Barkey. If you consider this this expression that is minus one toe the power key X to the power key. We can write this thing as minus one times X to the politics to the bar. Okay, that means minus X to the power key. So the series becomes Sigma minus K to the Power K and K is from zero to infinity and they're do part of the question in the ape articulation. We have to evaluate f zero then if off 0.2 if off zero point Fife F off one and finally f off 1.5 Now to do this, we know one formula from the Terram 10.7 that the CDs, which is Sigma, equals 20 to infinite. Hey, times are the power key is equals toe. A divide by one minus are on this form that is valid. Only win are more off. Art is less than one. If you consider that given cities with the standard one. We can see that in our case, it is one and our is minus six. Now we can evaluate all dysfunction. That means F zero means we have to find the value of the function at X equal to zero. So it will be equals two. A developer, one minutes are so in our case, here is 11 divide by one minus are here is minus six. So minus and minus will become place and X in this case is zero. So one divide by one which comes out to be as one. Next, we have to evaluate F off 0.2 So following the same procedure will can write one divide by one plus six on it will be called So one divided by one place 0.2, which comes out to be as nearly 0.833 Next, we're going to evaluate F off 0.5 So falling the same procedure one divide by one plastics on here x 0.5 So one plus 0.5, which comes out to be as 0.667 Next, we have to evaluate f of 1.5 off 1.5. But F off one on F off 1.5 does not exist because in this case, the condition which is moored off X less than one is not a valid. So that is why this function doesn't exist. And so this little answer for the A part and in the be part of we have to tell the domain of the function. So the domain off the function is minus one comma one.

Main event for actually looking for the interval of convergence. So quite used to 50. It actually used to test to figure that out here. So we do K root of X. Two. K over 32 Okay. She called subsidy bucks for three. Just less than one, which means the absolute value of X. Less than three. So we have potentially from negative three 23 Test the boundaries X equals -3. -1 to the K just diverges At x equals three. You get the submission of one. It's also courageous. So that means that the domain It's going to be from negative 323 stands for the party. Second part B Re evaluating half of two and half of dated one? Well f of two is equivalent to 2/3. Okay power. So that's equal to 1/1 -2/3. That's because that's where K equals 0 to infinity. So we have one over 1/3. She goes three. Next plug in negative. Once we have negative 1/3 to the K submission from Can equal 0 to infinity equals 1/1 plus one third Articles. one over 4/3. She goes 3/4. Yeah, that's the answer to part B. So for part C, we're looking for what F of X is general. That's next to the over 3 to the K power. She looks 1/1 -6/3. Playing top bottom base three here gives us three older three minus X. So this is our half of max function.

Probably need to find the domain of affects and preventing internal convergence. So let's go ahead and take the limit. SK approaches infinity imply the root test here. So we have experienced here. Okay, over to to the K That equals the value of exports to over two and It's supposed to be less than one. That means we have the absolute value Of X -2. He listed to thanks is to as being between negative two and two and two. To both sides. We have zero is the next less than four. And if we plug in next equal cereal here, you can see that we get just negative one. Okay, So which diverges that at X equals for We get the submission of one to come here. So which also to purchase. Okay, something from here. That means the domain This from cereal to four. Okay, assistance to depart party. OK. for F of one. It's equal to the summation of to get going to the K over to take a So that's equal to a negative one over to to the cape power here, that equals 1/1 plus one half. She goes 1/3 over to she goes 2/3 Koreans The next one which is up of two. She goes to get it to you. So, so that leaves us with so it's going to be negative too. Okay, over. Yes. Hard to to the K. So that is just negative one to the kid here, submission of here. So be 1/1 plus one Jay Equals 1/2. So then Okay, on actuality, so it's actually two minus two here. Which leaves us with had floor with a serial over to the K. So that will leave us with the zero here and which means that our answer here is just what? Yeah, it's the effort to. And then so if we wanted to find out what F of X is here, summation of Thanks -2/2. Okay, power here. So plugging that in, we have 1/1 minus Next minutes to over two. Bye Bye. two or 2. Top and bottom, we're left with 2/2 minus experience, too. That's simplifying this here. That equals to over four minus X. And that's our Alpha Becks.