Original subtitles

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Today we're talking about how to find the values for which the geometric series is convergent or converges.

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And in this particular problem we've been given the infinite sum from N equals 1 to infinity of the

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series negative five raised to the power times x rays to the power.

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Now we wouldn't know necessarily unless we were told that this was a geometric series so in order to

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prove first that it is a geometric series all we want to do is write out the first couple of terms of

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the series so that we can make sure that it is in fact in the form here that we have for a geometric

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series which is a times the quantity 1 plus Arpels are squared plus our cubed where r is that constant

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multiples so the first thing we want to do is recognize that in this particular problem this series

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that we've been given we have two values that are raised to the same exponent.

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When that's the case what we can do is actually combine the bases and just raise them both to the end

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power.

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So this series now becomes instead of these two separate terms here becomes negative 5 x all raised

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to the power.

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That simplifies our series A little bit.

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So now when it comes to writing out the first several terms of the series what we want to do is start

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plugging in an equals one equals two.

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The reason we start with N equals 1 is because we're told to write here because we have any equals one.

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So we start with N equals 1 and we get negative 5 x rays to the first power when we plug in an equals

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two we get plus negative 5 x raised to the power of 2 if we keep going we'll get negative 5 x to the

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third and then we'll get negative 5 x to the fourth etc. and we could continue on but you get the idea.

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So now can we get this series in the form here.

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Eight times one plus etc..

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Well we recognize that we need what is inside the parentheses here to start with one.

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That means that in order to do that we need to factor out the first term entirely so knows this series

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here starts with a we factored out a.

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And that just left us with this series starting with 1.

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So we're going to factor out the first term entirely and we're going to say here we're going to factor

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out negative 5 x.

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And when we do that when we factor that out we're doing it negative 5 x times 1 because negative 5 x

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times 1 gives us this negative 5 x to the first power here.

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Then we're going to get plus negative 5 x because negative 5 x times negative 5 x gives us this quantity

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negative 5 x square.

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Then we're going to get plus negative 5 x squared because we have here negative 5 x cubed.

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We factored out one of them negative 5 x times negative 5 x squared gives us this quantity negative

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5 x cubed.

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So we could keep going here.

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Negative 5 x cubed Plus the dot dot.

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But you get the idea there.

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And what we see now is that we have we do in fact have a geometric series in the form that we've been

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given here.

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So we have this value here of a.

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In our case our value of a the value we factored out to get our first term was this negative 5 x.

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Now we have the rest of the series here starting with 1 and now we have a constant multipole this value

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of our value here of our.

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In our case that's this negative 5 x value and we can see that we do in fact for our next term have

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r squared and then r cubed r the fourth etc. which is the form we know that are geometric series is

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going to take.

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So we now know that we have a geometric series because we've matched to this forum and we know the value

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of a and are at this point we have everything we need to determine where the series is convergent.

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We know by the geometric series test that the series will be convergent where the absolute value of

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r is less than 1.

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The series is divergent where ever the absolute value of r is greater than or equal to one that's the

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geometric series test for convergence.

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We know that our value of r is negative 5 x.

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So that means that our series is going to be convergent.

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Wherever the absolute value of negative 5 x is less than 1 so now it's just a matter of solving this

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inequality to find out where our series is convergent.

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So the first thing we want to do is go ahead and take away this negative sign because these absolute

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value bars are going to get rid of this negative sign automatically for us anyway.

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So we're just left with the absolute value of 5 x is less than 1.

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Now in order to remove the absolute value bars we actually need to add to this inequality that 5 x has

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to be greater than negative 1.

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So negative 1 less than 5 x less than positive one.

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That's what we have to do when we take away the absolute value bars that we have to do it.

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And then in order to get a value for x we want to solve for a value for x.

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We just divide through each Pease's inequality by 5 to get x by itself.

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What we're left with here is negative one fifth less than X..

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Less than positive.

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One fifth and we can see that the values for which the series is convergent are all values where X is

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between negative one fifth and positive one fifth.

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In other words greater than negative one fifth and same time less than positive one fifth.

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So within that interval the series is convergent outside that interval if X is less than negative one

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fifth or if x is greater than positive one fifth then we know on those intervals the series will be

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divergent.

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But within this small interval here the series is convergent.