In Exercises 37–44, find the sum of each infinite geometric serie...

Question

In Exercises 37–44, find the sum of each infinite geometric series.

1 - 1/2 + 1/4 - 1/8 + ...

Accepted Answer

Hello. Everyone in this video. We're going to be looking at this practice problem where we're given the infinite series one minus 1/5 plus 1/25 minus one over 125 and so on. And we want to evaluate the some of that series. So the first thing that we're going to want to do is figure out what sort of series we have. And in this case you can see that we're alternating between negative and positive values. So that's an indication that we're multiplying by some number that is negative, which is causing our signs to alternate. So that means that we're working with a geometric sequence. And when we're working with a geometric sequence, one of the first steps is to identify the common ratio and I'll denote the common ratio as our So in order to find our I'm going to divide The second term negative 1 5th divided by the first term which is one, so negative 1/5 divided by one is negative 1/5. So that tells me that my ratio is negative 1/5. So when I multiply my second term negative 1/ by this ratio I should get my third term. So now I have negative one times negative one is positive one and five times five is 25. So this indeed is my third term. So my ratio is good and works out for both my 1st and 2nd and 3rd terms. So I can assume that my this is the common ratio. And because we're working with a geometric sequence and we want to evaluate the some we can use the infinite sum formula for geometric sequences which which states that the infinite sum is equal to the first term, A one divided by one minus the common ratio are so I know all of these values So I can go ahead and plug in. So my first term is one and that's being divided by one minus my common ratio which is negative 1/5. If I simplify my denominator I have one divided by one minus. So the two negatives cancel out. So I have one plus 1/5. And in order to make this easier to simplify, I'm going to rewrite the one in my denominator as 5-5. So this becomes one divided by six over five. And I'm going to move this division into a linear format to make it easier to simplify. So I have my first term, the numerator one which I'll write as 1/1 and that's being divided by 6/5. And to change this into a multiplication, I'm going to switch the numerator and denominator of my second fraction. So now I have one divided by one times 5/6, which means that my infinite sum is equal to one times five which is five and one times six which is six or 56. And this becomes my final answer for this question. And you can see that this aligns with answer choice D. So hopefully this video hoped and see you next time.

In Exercises 37–44, find the sum of each infinite geometric series. 1 - 1/2 + 1/4 - 1/8 + ...

Key Concepts

Infinite Geometric Series

Sum of an Infinite Geometric Series

Common Ratio

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