Find the sum of each infinite geometric series.

Question

\sum_{i = 1}^{\infty} 8 (- 0.3)^{i - 1}

Accepted Answer

Hello, everyone in this video, we're gonna be looking at this practice problem where we are given the infinite series, the sum of six times negative 0.4 to the power of J minus one. And we want to evaluate the sum of that series. So in order to evaluate the sum, let's go ahead and take a look at some of the terms. So I'm gonna say J is the term that we are on and I need to plug in different values into J. So for example, when I have my first term J is equal to one, which means the first number in my series is six times negative 0.4 to the power of one minus one. If I simplify this, I have six times zero times negative 0.4 to the power of zero. So this becomes six times one or just six. If I look at when J is equal to two or my second term, I have six times negative 0.4 to the power of two minus one. So that's equal to six times negative 0.4 to the power one. So that leaves me with six times negative 0.4. If I look at J is equal to three or the third term, I have six times negative 0.4 to the power of three minus one or six times negative 0.4 squared. So between these terms, you can see that I'm multiplying negative 0.4 between terms. And because I'm multiplying this ratio, I have a geometric sequence. So when we're working with geometric sequences, it's important to find the common ratio. And in this case, you can kind of see that it's negative 0.4. But we can confirm by dividing the second term six times negative 0.4 by the first term six. And I'm gonna denote the common ratio as R. And in this fraction, you can see that the six is cancel up to one. So I'm left with the common ratio of negative 0.4. So now that I have the common ratio and I've determined that this sequence is a geometric sequence. I can use the infinite sum formula of a geometric sequence to determine the sum of this series. So the infinite sum formula for a geometric sequence is a one or the first term divided by one minus the common ratio R. And I have all of these values because I solved for the first term. So a one is equal to the first term which is six and I have the value of R right here with, with it being negative 0.4. So I can plug in to this equation. So I have a one. So six divided by one minus negative 0.4. If I simplify my denominator, I have six divided by one, two negatives make a positive. So I have plus 0.4 or six divided by 1.4. In order to write this as a fraction with no decimal points, let's change 1.4 into a fraction. So 1.4 is the same as the mixed number one and 4/10. And if I make this one into a fraction, this would be 10/10. So I have 10/10 plus 4/10 which means that 1.4 as a fraction is 14/10. So I can go ahead and plug that back into my infinite sum here. So I have six divided by 14/10. And because this 10 is in the denominator of my already already divided six, I can move it in front to be a multiplication. This is the same as saying six, divided by one is being divided by 14, divided by 10. And we change this into a multiplication by switching the denominator and numerator. So my second fraction becomes 10/14. And here you can see this is equal to 60/14, which I can simplify by dividing by two. So 60/14 is the same as 30/7. So my infinite sum is equal to 30/7. And this becomes my final answer. And you can see that this aligns with answer choice B So hopefully this video helped and see you next time.

Find the sum of each infinite geometric series. $\sum_{i = 1}^{\infty} 8 (- 0.3)^{i - 1}$

Key Concepts

Infinite Geometric Series

Common Ratio

Sum Formula for Infinite Geometric Series

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